How do you cook more successfully in a different kitchen?

Assuming you are not catering, and this, as you state, is most often holiday meals with friends and family, then practically speaking, you can bring your own knife (or knives), and a positive attitude. What else can you really control? Maybe, during these times, the ideal food is not as important as the interactions of family and friends.

If you're vaguely in charge of the menu, you could cook things with non-crtitical timings.

Nothing worse than one pan going off like a shot while the next one seemingly takes forever to come to the boil. You're thinking, "why can't these people have sharp knives, pans & rings that all behave the same… spices where I want them to be…" Then you realise their oven runs at least 20° cooler than yours…
Your external appearance is one of calm, but inside you are in a flat panic.

You can't beat that, you have to relax & enjoy the day. Non-critical timings will help that a lot.

Late addition
If you can't control what's being cooked, try to control what part you play - do prep - sprouts for 15 will keep you nicely occupied & looking helpful for a while, or take charge of anything long-cook where you can, something set-and-forget. Anything that means you don't also have to figure out what their cooker is like.

Also… You can sharpen their cheap, soft knives on the spine of another. It will scar the spine of one knife & the edge won't be great, but it will cut… & they obviously didn't care about their knives anyway ;)

Personally, I do a few things:

1. I keep a bin of supplies in my car. It's more geared towards serving food rather than cooking initially (as I'm mostly dealing with meetings where there's food being served, and they don't have the proper supplies). Although it has paper goods (napkins, paper towels, plates, bowls) and plastic utensils, I also have serving tools (smaller tongs & large spoons), cloth kitchen towels, an apron**, knives (chefs, paring, and serrated), a metal spatula, cutting mats, a probe thermometer, a pair of 12" metal tongs, salt & pepper.

   It's actually a bit of a joke with my friends & family, as I'll often say "I think I have (something) in my car", and can then produce a straw, plastic table cloth, or something else unexpected.

   My bin is actually two restaurant bus bins stacked together, with a lid. So I have a second bin available to collect up all of the dirty stuff to be cleaned.

2. For people whose homes I cook at regularly, I give them gifts of cooking tools over the years ... house warming, christmas, birthdays, etc. So then I know that they have some basic stuff there for me to use.

3. If I know that I'm going to be going over there to cook something specific, then I'll either grab one of my other bins of supplies (I have three -- one with grilling supplies (longer tongs, long handled spatulas, grill brush, welding gloves, infrared thermometer, more aprons, and space for m to throw some aluminum foil in there, etc); one for tea / coffee service (hot cups, lids, stirrers, sugar canister, artificial sweeteners, assorted teas); and another with cleaning supplies, deli & takeout containers, ziploc bags and similar for cleaning & packing up food).

   Or I'll throw together a new bin of things that I think we might need for that specific event. I try to ask ahead to figure out what they're less likely to have, and focus on those 'nice to have' things like microplane zester, mandoline (with a cut-resistant glove!), larger pots, but will also try to make sure we have duplicate things like vegetable peelers, knives and cutting boards, so we can have multiple people working if there's sufficient space.

   Depending on the where I'm cooking, and what the item is, this might turn into a situation like #2, where I just intentionally leave items with them for next time. (I've given out a lot of sheet pans through the years, as I had access to restaurant supply stores)

Now, when it comes time to actually cooking in someone else's place

• As you mentioned, don't trust their oven behaves like theirs -- check things early, and adjust as needed. I've also run into strange cases like a burner that was either broken or only intended for simmering -- it just refused to boil a large pot of water.

• Give yourself extra time. Even if things don't go horribly wrong, things are unlikely to be like your most optimistic time predictions and you'll want time to recover.

• Try to have someone who lives there hang around in the kitchen. Even if you're doing all of the cooking, they can be getting out supplies for you, so you're not spending 5 minutes going through all of the drawers trying to find a fork.

• Be flexible and creative. Although it drives one of my friends crazy, you just need something that works, even if it's not ideal. So if I'll mix sauces or scramble eggs in a (clean) coffee cup with a fork if that's what I know where they are or find first.

• Try to take inventory of what you have before you start. It really sucks to think 'oh, it's still in the car' only to find out that you forgot to pack something, or something that you thought they have they no longer do. If you take a quick inventory before you start, you can send someone to the store, or try to call someone who's still on the way. (for my office's annual "awards picnic", we would specifically have a couple of people held in reserve for this, as the location took a while to get to).

** It's usually one that says "Don't Make Me Poison Your Food", but sometimes it's "The Spice Must Flow" or some of my even less appropriate ones.

In addition to things like knives that others have mentioned, I have brought things like induction cooktops (with pans), instant pots, crock pots, roaster ovens, etc. That's often helpful because you just need to find your own corner to plug in, rather than coordinating the use of a shared oven or cooktop.

For ingredients, it can be helpful to go more mise en place than you usually do at home. Get everything you need together first, so you are just quickly dumping things in, instead of having a deadline for finding the next ingredient.

I interpret "Help in the kitchen" as meaning that you are not the primary cook, so you don't completely decide what to cook.

I'd bring in a few items that could be useful and are likely to in short supply, if they're even available at all:

• As others have suggested, a few good knives. Cutting board? Might be too big, but it could be useful.

• Probably some measuring utensils. My breadmaking machine has a gradated plastic measuring cup that is lightweight, for example.

• Spices/herbs. Say you're helping with desserts. Having cinnamon, nutmeg, cardamom in small quantities quickly allows you to chip in some extra flavors. Meat? - how about unexpected things like tamarind paste or smoked chipotle? Fresh herbs are also a possibility.

• if you are cooking a dish yourself, figure out the exact recipe you want to use, print it out. Check with the host what ingredients are available there.

(TLDR version: Become a leader. The rest is just adaptation and experience.)

Great answers already about bringing your own knives (buy a knife roll/bag) and maybe even extra things, especially for the dishes you plan to make. I've also stocked up at least 3 people's kitchens with things they didn't have because I cooked there frequently enough. Yeah, I'm out a few hundred bucks, but my holiday stress level is reduced. By that same token, try to bring anything that's important for your dishes, especially if you're not sure about what the host keeps around -- your set of spices/herbs is a big thing is often forgotten when packing your ingredients.

However, the biggest advice I can give you is to become the respected cooking expert in the group. If you take a serious role in helping plan and execute the major part of the meal, just letting others volunteer to bring pre-prepared dishes -- i.e. they won't be in the kitchen -- and you become the "problem solver" for the cooking, over a couple of years, you will start to take on the role of the kitchen leader, and others will defer to you. Pretty much every kitchen you walk into where the meal was planned with you will become your kitchen for the day. If you're lucky enough to have someone in-the-know with a good sarcastic streak helping, you might even get the occasional mocking "YES, CHEF!" to let you know when you're getting a little too bossy and forgetting to say "please" and "thank you".

Note that you don't get here through a hissy fit or power play. This requires actual leadership skill on top of having a passion for cooking and being willing to take responsibility for the success of the meal. It also means you're taking on the stress of it, but you're doing it because you like it. This is exactly why this method works; people like to follow a confident leader and let them take responsibility. If someone else is running the kitchen, then just remember that it's their stress, not yours, and you're just there to help them as best you can, so don't sweat the small stuff.

Once you've moved into the kitchen leader role, you'll probably find that people will not just invite themselves into your kitchen (that's rude). They will volunteer with "is there anything I can help with?" Since you should now be aware of everything going on in your kitchen and what needs to happen, you should always be prepared to answer this question. If there's nothing at the moment, then defer to the host to see if they need anything done not in the kitchen. But also realize that now you can control the kitchen traffic. For instance, "Yeah, actually, I could really use someone to prep these Brussels sprouts. Here's a paring knife and a couple of bowls; there's more space at the table if you want to sit there and do it." It's also OK to say "not at the moment, but check with me again in 20 minutes".

And as Joe mentioned, try to keep someone who lives there nearby to help find things. They may be your "sous" for the day, or they may not even be much of a cook themselves. I would go a step further by pre-gaming the cook with your main "crew" and asking for the location of major items you will need. If there's room, maybe even do a little supply mise en place so you can just go to your "supply station" instead of remembering. Call out for things while you're finishing tasks and let your "assistant(s)" locate them for you. Do all your sink-centric prep work early, then let a volunteer help clean dishes and clear away the mess while you and the other cooks focus on cooking. You'll definitely need that big bowl again, and someone at the sink won't be in the way too much.

Source: Experience. I run every home kitchen I walk into now by default. It helps that I'm also comfortable in a leadership role in general, so this is not just a kitchen thing for me. If I'm not the head cook for the day, I'll drop by the kitchen as I'm unloading anything I brought and just let them know, "Hey guys, I'm here, give me a shout if you need anything." Usually, I just enjoy myself and let them work. Frequently, I will be called in to taste-test, so at most it's just suggesting a pinch more salt or a splash of vinegar to brighten the gravy, or saying "I think it's good!". Sometimes I may get called in to troubleshoot, at which point I usually become the de facto leader again. This means I need to be conscious of the whole kitchen, especially because the overwhelmed head cook is now worried about this failing dish. So if it's taking a while, I may have to say, "I've got this for now if you need to check on your other food" to remind them. Once I'm done helping with that, I step out and let the cook take over again. I'll come back and check on them to make sure they're still good, but otherwise stay out of the way unless they request me. If I do get asked to stick around, I often have to remind them that it's their meal and I'm just helping, because people will instinctually expect me to take charge. So be aware of that, too -- once a leader, always a leader, even when you're in the follower role that day.