QUESTION 1:
How do I take snapshots, both in battles and while in the hangar, without the huds? Is there a way to easily make the huds invisible?
Alt + H while in battle 
Not sure about in hangar
most in hangar shots are made with “preview”, since that turns off most of the overlays, and leaves only the interface and the name of the ship visible. you can preview any ship.
Thank you for the previous answers.
QUESTION 2:
Are the mission given by Contacts (R. Fris, S. Galo, etc.) compulsory to complete?
if you mean the missions where you send your ships away for a few days?
no. they are not part of the 33 basis missions, or the daily alien monocrystal missions. no mission is compulsory, but i would do the 33 missions per faction, and the daily invasion missions if you have time.
Thank you.
The first thing you need to think about doing as much as possible are the contracts for each faction, in order to get Loyalty so that you can upgrade your ship modules to MK3. Those are done by themselves just by being aligned with a faction while playing PvP matches. Some contracts you can do by PvE also. After you’ve finished or mostly finished with a faction, you should change your base in order to do loyalty with another faction.
Then you should also do the daily crystal missions in invasion if you have enough patience, because it’s the only way to get Alien Monocrystals. You will need about 450 of those in order to craft a special project ship (the ones at the end of the ship trees). And you can get only 6/day without premium, 9/day with premium. So yeah, if you plan on having those ships (they are nice because you can arrange the slots however you wish when you craft it, but make sure you pick a good combo otherwise you’ll loose a whole bunch of crystals), you should start getting at least a couple of monocrystals/day. Will take a while, this is for the most dedicated players. It’s not a must.
Thank you for the advice.