What kinds of training and research are possible within the paradises found within a Rod of Security?

You have a paradise limited only to your imagination, but this is more of a constraint than it seems.

The description of the paradise is very open ended, but that doesn't mean we can't define how a character could reasonably improve while inside. First, some housekeeping:

• What you can imagine:

There are only two constraints on the paradise: you have to be able to imagine it, and that it is a paradise. It's hard to define what is and isn't a paradise, but it's reasonable to say that a paradise doesn't contain anything that could mortally wound you, like an active volcano or tarrasque. The nest of an ancient red dragon isn't a paradise by most people's standards. It's important to bring this up, because if you could create a dragon, you could kill it and gain the XP for defeating it. Things inside the paradise aren't illusory, they just can't be taken out of the paradise. Since the Rod doesn't say you can't have creatures appear in the pocket dimension, a player could populate their paradise with creatures as long as they didn't make it not a paradise. There's an argument here about having peaceful creatures appear, but it's reasonable to say that any creature that can harm you wouldn't appear in a paradise (but what if I imagined an ancient red dragon who had sworn a vow of pacifism? That's another question, though).

• What you can't imagine:

The only thing that you (or your character, more accurately) can't imagine is something that they don't know or haven't seen before. You could imagine a library, but it's unreasonable to "imagine" a book with information you've never learned before. That rules out studying in a library, or anything else that would require you to learn from something that would appear in the paradise (i.e. "My paradise contains the secret incantation to dispelling the lich's phylactery"). There's an interesting quandary here for druids - you could imagine a fantastical beast with adamantine talons, a flying & burrow speed, and truesight, even if that creature doesn't actually exist. Could that druid then wildshape into that creature after seeing it?

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We'll use what's contained in the above categories as our constraints as to what the paradise can have in it. No hostile or potentially dangerous environments or monsters, and nothing you don't already know. We'll go down the list of possible training, downtime activities, and other forms of character improvement with these constraints in mind. We'll also use the downtime rules from Xanathar's Guide to Everything, as they're more robust.

1. Levelling up: We've already established that you can't create a creature with the capacity to harm or kill you in the paradise. That means you can't get XP by killing monsters, but your DM might use milestone levelling or award non-combat XP for social encounters that might occur in the paradise. You can gain XP, just not by killing monsters. Levelling up is possible.

2. Training (tool proficiencies or languages): We've gone over how you can't create information in the paradise, but there's nothing preventing you from bringing books and instruction manuals into the paradise yourself. In the same way, you couldn't create a teacher in the paradise, but you could bring one with you. The maximum amount of time it can take to learn a new language or tool is 10 weeks, with it taking less time for characters with a higher intelligence modifier. 10 weeks is 70 days, so you and one other creature (your teacher, presumably) could enter the paradise and have time enough to learn something new.

To answer your questions more specifically:

1. Gaining new tool proficiencies: Yes! As long as you're bringing a teacher or some other source of learning in with you, there's no reason you couldn't learn a new proficiency or language. For tools, you wouldn't even need to have a tool kit, as long as you could imagine one to use while learning.
2. Catching up on downtime stuff: We'll use XGtE downtime activities (XGtE p.123) as our examples. Some of these activities involve interacting with the outside world, such as selling a magic item or carousing. Unless you're selling a magic item to or carousing with creatures also in the paradise, you couldn't do these. Others, like research, crafting an item, or scribing a spell scroll, are entirely plausible provided you bring in everything you would need to craft something and any material you would need to learn from. The "Relaxation" activity is the best example of a downtime activity you could partake in - you're in paradise, baby! Kick your feet up on that pearly beach and have a feast twice a day for a month.
3. Research old &/or new spells: We've established that it's possible to level up inside the paradise. If a character gained a level in a casting class, there isn't anything stopping them from adding to their spell list like normal. I'm not sure what you mean by "old" spells, but if a wizard brought a spell scroll into the paradise (along with enough special inks and materials, PHB p. 114) they could still copy that spell into their spellbook.
4. Develop a formula/blueprint/pattern for almost any magic item: This falls easily into the activities of "creating a magic item" and "research". There isn't anything saying that the pocket dimension functions any differently from the material plane, so any research that you perform or magic item you design has no inherent reason to work differently outside of the paradise. As long as you bring in any academic materials you would need to learn from, and any magical components you would need to build a magic item, you're set.
5. Research ancient lore in a paradise-library: You could create a library paradise without a hitch - leather armchairs and tall, cozy shelves as far as the eye could see - but as we've gone over above, those books would have to contain information that your character already knew, since they can't imagine something they don't know or have never seen before. As long as the character is bringing the ancient lore in with them, there's no problem!

To conclude, yes, the Rod of Security is a powerful magic item. But when you look closer, it can't provide the characters with anything more than a guarantee of safety for a limited time. It can't give them anything they haven't seen before or teach them something they couldn't have otherwise learned.

TL;DR: The paradise can't contain anything that the players could kill to level up, but they could gain XP in other ways. The paradise also can't provide the players with knowledge the creator didn't have already, but a character could research or design something as long as they brought the subject of that research in with them. Any downtime activities that would require interacting with creatures or objects outside of the paradise also can't be done.