Does it make sense to make claims when I not yet intend to declare war?
I am currently surrounded by multiple "friendly" and "cordial" empires who have not yet realized that ultimately their conquest through me is inevitable. But it might still take a few decades until I will start my galactic conquest.
However, I noticed a new game mechanic in Update 2.0: I can "Make Claims" on their territory, which costs me Influence but apparently is a prerequisite for claiming their worlds when I declare war on them.
Does it make sense to make those claims now or should I wait until I am about to declare war? What happens if they declare war first and I don't have any claims yet? Can I still make them or am I out of luck? And will making those claims hurt my relation with them?
Claims cost influence, not unity. Having claims puts a negative opinion modifier towards your diplomacy with that empire. You need at least some claims to start a war of conquest. Making new claims during a war is more expensive.
So the overall strategy follows: stockpile influence, make claims during peace when your are nearly capped on influence, claim all you want right before you declare war.
Notice that multiple war participants can claim the same province. If you notice someone tries to claim your target, you might need to claim the same system multiple times to guarantee that it is you who takes the system in a peace deal.
Update following the discussion in the comments:
You can also gain territory if you fight well in a defensive war. For that you use the same claim mechanics. You can create new claims even while defending. Fight the war (both defensive and offensive) up to the moment when you're satisfied with the Status Quo peace, and grab all claimed systems, occupied by you, and lose all systems claimed and occupied by your enemy.
This allows you to expand even while fighting defensively or attacking with some casus belli other than conquest.