Now that some time has passed and I am more or less familiar with my Raspberry Pi, do you have any suggestions on what I could do with the given hardware? I mean something semi-useful, not just pretty blinking patterns...

Do you want to do something where you're interfacing it with other hardware, or are you looking for more of a software project, such as using it as a server of some sort?

Well, actually I am open to any ideas...
Both would be fun. But I mean something with the given hardware, which makes my question harder to answer. (I mean, just any Raspberry Pi project would be a question for Google, I am looking for... uh, more custom projects.)

Did you have anything particular in mind, for no matter which of the two categories?

Looks like I'll put my RPi to use, after all.
We've done some brainstorming with a friend at school and thought that having a portable minicomputer would be neat. There were three suggestions on the possible appearance of it:
1) Use an old book as a case, remove all the pages and add some cardboard or paper to make a box from it. The lid would be the book cover itself. The interface would be below the lid, anything inside would be be below the lid and any sockets would be on the sides.
2) 3D-Print a case (We do have 3D printers at school). I don't think any further explanation is needed.
3) Least likely to be done, neither of us really liked the idea: Make a box similar to the book one, but from wood.

Looks like my friend will end up with 1) and I will end up with 2), should we both decide to build a minicomputer.
These were the only things that we didn't settle upon. In general we both agreed that we would use touchscreens, mini keyboards, no mouse, a powerbank for a supply of power and, well, that's it.
A problem would be, at least in the book design, how to manage the cooling.

Any thoughts, ideas or suggestions? This is only an ambitious project idea that may come to life in the summer vacations or so, and would require at least some guidance by experienced people, and that's why I am positing it here.

Thanks!

I really like the idea of hollowing out a book to use as a case for the computer, with bonus points for how much you can make the final book still look like the original book. 3D printing a case would be cool too, but not as original as the book case.I bet you could be clever with cutting out the edges of some of the pages to make vents for the cooling problem.

This was indeed the plan. Also, you gave me the idea to make a vent by cutting off part of the edges on two pages, then two intact pages, two cut out pages and so on... Should be hard to see and still more or less a vent.
However this would require indeed hollowing out the book. While the plan was to remove the pages completely and just replace the walls with stripes of paper, which is easier, lighter and, perhaps most importantly, insulates the electronics the least. As I could imagine that a thick layer of paper (1 cm? 0.5 cm? I doubt less.) insulates pretty well.

I figured out that a 5V/5000mAh powerbank will keep my RPi & Display configuration powered for 5 hours if idle and 3 hours at medium CPU load. So I am now looking for a nice powerbank with 5000 - 10000 mAh capacity, however there are more things that are essential.
1) I'd like the power in and power out ports to be located on different sides of it - the same side would do too, but is inconvenient.
2) Some/Many powerbanks cannot power anything while being charged simultaneously - I'd prefer one which can do that.
3) Some/Many powerbanks require a button to be pressed to power anything - total no-go, I am not opening the whole case to press a button to start my RPi.
4) I noticed the powerbanks can be classified as rated for 500 charge-discharge cycles and rated for 1000 cycles. While non-essential, I would still prefer the latter type.
5) I am not buying anything that is supposed to hold 5000 mAh but in the end holds less than half of that. Total no-go again. I rely on reviews for that.
6) It should be of a fitting shape - I have about 6 cm * 4 cm * 15 cm of space available
7) It should not weigh more than 250 grams, aluminium ones are too heavy. I am not carrying a bag of aluminium with me.

These really narrow down my search. In fact I haven't found anything yet. Do you have any advice?

While my question from above stands, I have a new one still. I arranged to print a case at my school's 3D printer (Yay to good relations with teachers!) but I am really uncomfortable with requesting too much, as the teacher said it costs him a lot of time too. I would else make a mini test print of my model, but I would rather not, under those circumstances. So I figured, or rather I hope, there must be some software to evaluate the quality of a 3D model and "simulate" a print, also to simulate how more than one part can work together once printed. Is there something you know and could recommend for these purposes?

Edit: Note I found plenty of software to simulate a stress test of a single part, but nothing more. And even for this I would like to hear if you know anything that does simulate well, as the reviews I usually see for that kind of programs are not the best.

If you load up a model in standard 3D printing software like Repetier Host, then slice it, you'll be able to see all of the plastic that the printer is going to print, including the raft, brim/skirt, and supports. Unfortunately, there's no great way to know if it's going to be a successful print without a bit of experience. You want to make sure that it's not going to be printing anything in midair, first and foremost, and that the amount of plastic used looks like a sane number.

That's not really going to help you much with 3D printing, I'm afraid.