3.20 is the golden firmware for a PSTV. If you got that FW, I suggest staying on that 3.20 firmware.
Wont it make me update when I link my PSN account to it?
No, not if you just link your PSN account at first-time setup. You WILL be prompted to update if you try to
activate your PSTV though. (basically activating the PSTV allows you to play content from a memory card, whereas if you still have the PSTV deactivated, you can't play anything from a memcard... Which pretty much sucks in my opinion.)
However, you CAN bypass the update nag when trying to activate your PSTV. But it
does require a PS3! All you have to do is transfer some content from a PS3 to the PSTV, be it a PSP game, a Vita game, basically anything transferrable from a PS3 to a Vita/PSTV. Then it will automatically activate the PSTV and you should be good to go with playing the PSP game(s) from the memory card! (You will need the same PSN account tied to both devices for it to work though.)
Wont it make me update before I can access the PS Shop to download a PSP game for part 3?
Yes. But like I said above, just buy something from the PS Store on the PS3, and transfer it to the PSTV, then you are ready to start the tutorial.
Is there a restriction to what game can be used for part 3?
I don't think so. Just make sure it's purely PSP content.
And I assume you cant keep the game you use for this, right? (Not that big of a problem to be honest)
Actually, you can still run the replaced game from within TN-V or ARK afterwards!
Can you run PS1 games (Using the old Popstation exploit, I believe) using either hack (VHBL or TN)?
You're best option for PS1 games is TN-X (Requires a legitimate purchase of Tekken 2, if I remember correctly). TN-X is completely different to TN-V or ARK, but it's meant to be really good!
It also mentions that you can update to 3.55 and keep VHBL / TN, is this true for 3.6 as well? Is there any benefit of keeping an old Firmware aside from the whitelist?
3.57 and 3.60 patch the current PSP-side kernel exploits (TN-V and ARK), so no. Also, updating to 3.55 would be pointless, because it patches out nearly all possible Vita/PSTV exploits (including whitelist), and still doesn't allow you to access online/PSN content.
Can I log into a PSN account after installing VHBL?
Nope. VHBL is homebrew-related anyway. There's just no way to enable PSN content under the latest firmware yet. :/