Are missions worth completing even if it means opening up a hole in my defenses at the start of the game?
It really depends on what difficulty you're playing against. If you're playing against anything other than Cruel or Vicious, it's very simple to set up a starbase or two at choke points and watch your other enemies suicide their entire fleet on it (if they attack you) - this allows your own fleet to go and try to complete a mission the AI you're interested in asked you to do.
It also depends on whether you'll have enough resources to get a cease-fire with the AI - it might be easier to complete a mission than to get the relationship bonus from some other source (like expensive research, or expensive donations to the AI). For some missions, it's relatively easy, even without doing anything: sometimes the mission will just be "Kill Ships: 5" and if the target enemy sends a few ships to attack your starbase, you can just sit back and watch and the mission will be successfully completed.
This is just a thought, but it might be possible to boost your relationship to reach that key point of a cease-fire by sending envoys to one of that AI's gravity wells - one with very few ships, or the envoy'll be shot down immediately. If you can, use the envoy's abilities to increase your relationship with the AI.
However, if the AI's main battle fleet arrives, the envoy will be in serious trouble: if this happens, you should have a couple envoys in reserve, ready to warp in to the AI's gravity well at a moment's notice. When the first envoy starts taking serious damage, activate the ability to make it invulnerable, and then order it to warp out a few seconds later while telling one of your other envoys to warp in. Kite around, repeat. If you have 4 or so envoys, you'll be able to keep this going almost indefinitely. (You always need to have an envoy at the gravity well or in route ASAP so that you don't lose the relationship bonus from the gravity well)
That's only an idea though, and requires a lot of attention - it would only be worthwhile if it's too difficult to get the AI to like you any other way.
These numbers are useful to know:
https://www.sinsofasolarempire.wikia.com/wiki/Simple_Diplomacy_Modification
especially the
"offerOrAcceptTreatyHappinessThreshold:CeaseFire" value of 0.5 - that's the magic number you need to get to to get a cease fire going so you can start sending envoys.
I remember once I was trying my best to get an AI to like me but wasn't able to quite reach that 0.5. In the end I had to scuttle one of my fully upgraded planets to remove the "Adjacent Territory" penalty, and I was finally able to get a cease fire, envoys, and then recolonized.