On the topic of autoresolve. I'm sure this is silly and already addressed but... is it possible for autoresolve to just happen to pick your command flagship (or whatever it's called) as a causality and lose you the game? It seems like it's been implied your main ship is a unit in the RTS side of things.
No, autoresolve is smarter than that. It gives you an overall danger level estimate, first of all, and you can be reasonably sure you will win if the danger levels are moderate or below (just with varying levels of casualties). The flagship of either side doesn't "enter combat" in autoresolve at all until all the ships on its side are already destroyed, so you don't have any sort of the subtleties of flanking or "the enemy's gate is down" sort of play. If you want that sort of thing, you have to play it yourself rather than auto-resolving. But on the plus side, you also don't have to worry about pure chance losing you your flagship while half your fleet is around.
Which is true also opens the question of, does it feel like you need to babysit the unit?
Not really. Think of it like a mobile town center. It moves incredibly slowly, although it does move. It is what is deploying your ships, so wherever it is is where you tend to orient yourself around. The only real reason to move this ship at all is if you want to put it behind some cover, or get a better angle on your enemy, or have it move further into or out of range. You don't generally do much of that sort of thing.
All of your ships immediately come out in "escort mode," where they just fly around your flagship, protecting it. So it's inherently passive on your side until you tell them to do something. You can put your ships into "fire at will mode," where they basically bumrush the enemies kind of like free roaming defender mode in AI War. Or you can select specific squadrons or just whatever arbitrary groups of ships, pull them off your main forces, and move them around to specific locations and then do whatever maneuvers you wish to do at that time.
Typically the way I play is to have some stuff on escort, then move around my mobile forces in one or more groups that I manually shift around. That's pretty easy to do, mechanically speaking. As I then react to the enemy AI, I either exploit weaknesses or draw my forces back, etc.
In other words, you do have to protect your flagship like you would your king in chess or your command stations in AI War. But there are almost no orders you have to give to it itself unless you want to tweak its positioning slightly (and very slowly).
