Connecting an arbitrary number of pins to pin holes without soldering?

As you already have seen, there are not many options to contact such boards without soldering, because they are meant to be soldered. Those hammer headers are an option. When you find a seller for your country, it should also have these in different sizes/numbers of pins.

Though contacting the board with the pins is not the only problem, you may face. As long as you are experimenting, the typical breadboard is often enough. But when you want to make something more permanent, reliable, current thirsty or noise sensitive, a breadboard is a bad choice. In that case you would also need to solder. Soldering is just a very integral part of electronics.

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As others have wrote in the comments, I would also consider soldering as a hobbyist as safe, especially, when you take some precautions. A "well ventilated area" can be a simple room with the window open and a small fan directed to the working place, so that the fumes don't directly go up your nose. That is already a very big improvement, which is also rather cheap. That's also about what the woman does in the video, that you linked. And washing your hands after soldering is also a good measure. (I'm doing soldering the described way for about 15 years now, without any health issues)

In the video, that you linked, they say, that the fumes are toxic and their effect adds up. Letting aside, that they don't back it up with any data and I cannot either, they also don't talk about the amount of solder work, that they consider here. As an electronics beginner you most likely will just solder very rarely for getting pin headers on boards or maybe contacting a motor. The main work most likely will still be done on the breadboard. So here the exposure is so small, that you don't need to consider more than the above safety measures. These will also last rather long, as you get more profound in electronics. And when keeping up that hobby, you maybe wanna invest in a more professional filter system, as you are doing more solder work.

Doing electronics while being that scared of soldering will not be much fun. For the start you might get away by only buying presoldered boards. But very fast you will run against walls with this. Not a good start for a beginner.

But in the end it's your own decision. You have all right to be afraid of solder fumes, though that just might disqualify electronics as a hobby for you.

if I installed hammer pin headers (seems like one has to hit it with an actual hammer to install it), can I un-install (remove) the headers?

That depends on the pin headers, but most likely no. You would need physical force to get them off. And the pads on a PCB are not very strong. They are made to be soldered, not for tearing the pins off without any damage. But why would you want to remove the headers? You only very rarely solder wires directly to the board for connecting with other parts. Either you solder it onto another PCB (where the pin headers come in handy), or you use a female header on a PCB or a cable for making the connection temporary. For example all my small Arduinos get pin headers soldered to them, so I can put them onto my own PCBs via female headers on the PCB. With that I can also disconnect or replace the Arduino if I need to.