So I am a part of this alliance where one empire is pathetically small and constantly votes against wars. I think it would make more sense for them to be vassals rather than allies, given their power. Is it possible to somehow achieve that the ally gets kicked out of the alliance? (I think if I leave the alliance the rest of the members still stay in the alliance so that doesn't seem to be an option). For example if I insult them and lower their opinion sufficiently will they leave on their own? I want to keep the other allies.
Patch 1.2 aka Asimov (which is currently available as a beta on Steam) has added the option for Alliances/Federations to be able to vote out a member. Any member can raise a vote to kick another member and they require half the membership + 1 for it to pass (e.g. alliance with 4 members would require 3 members to all vote to kick one member).
This can also happen to you, as other states may raise a vote to kick you if they have a negative opinion of you.
At the moment no. Alliances are more or less permanent arrangements. I don't think you are even able to reduce their opinion greatly, as multiple insults don't stack.
Paradox has announced to add more temporary alliances in future patches, but at the moment your only options are to leave the entire alliance or stick with the weak member.
You can work around the war demands by giving your allies liberate planets objectives and then peacefully vassalize the resulting empire.