Welcome to the Sins community.  It's a really great game.

Since it is a capital ship you might want to keep a close eye on it and just micromanage the use of its abilities.  Also, you could use it while it's out colonizing and fighting militias; it will help restore the shields on the frigates you brought along to help fight alongside your Mothership (along with the mothership's shields).  As a general rule, the AI does an awful job with using capital ship abilities, so that's something that you should plan to control directly.  (Actually, the AI does a poor job doing just about everything compared to what a human can do.)

Shield Restore is one of the most useful capital ship abilities in the game and part of what makes the Progenitor Mothership one of the game's best capital ships.  As Advent, it should almost always be your first capital ship.  Pros who play competitive online multiplayer use the Progenitor and the shield restore ability very heavily.  A common upgrade route for the Progenitor is:  Level 1--colonize.  Level 2--shield.  Level 3--colonize.  Level 4--shield.  Level 5--shield or colonize depending on whether it's time to fight or whether the cap will keep colonizing open planets.  (Exception-- if you play online multiplayer and you plan to use a scout/light frigate spam strategy and/or you anticipate that your opponent is going to try that against you then you might prefer a Halcyon carrier capital ship.  However, I still prefer the Mothership for my first capital ship.)  Shield restore is also very useful in combination with Guardians which can absorb 33% of the damage your other ships are taking and apply it to its shields.  (Mothership shield restore of your Guadians' shields is part of the Advent's synergy.)

Advent is a great race to specialize in, by the way.  An experienced and competent Advent opponent who has put together a "battle ball" is hard to stop.

I rarely have any abilities on autocast. The autocast AI just doesn't know how to use them. Shield regen should have the max amount of ships possible within the radius, for example. Same with Missile barrage, etc. The Advent microing of abilities gets pretty intense though. You pretty much are just using abilities constantly, late in a game.

One thing to keep in mind about Shield Restore - it is channeled, so if you tell it to do anything besides, it'll stop casting.

About hating micro - you should speak w/my best friend, who thinks in very much the same terms.

spefcial!!! Dey Spefcial!!!

Yes, it does, just not very intelligently.  It will usually wait for multiple nearby ships to be damaged before it does so.  I'm not surprised that it fails to do so just to save itself.

For most situations, I never turn on autocast.  Sometimes it's good enough, most often it does not.

Just last night I was playing a game, doing my standard fast expansion with a Progenitor.  It reached level 2 and I added my first point to shield restore.  Before I could turn off autocast (a reflex whenever I add to new abilities) it had already used the ability.  So yes, it does use the ability even if only to heal itself.

Admittedly it's not too difficult to line up your ducks and hit this ability every few seconds.

Shield regeneration is possibly one of the most powerful abilities in the game.  Its autocast AI is at very least better than malice, which will waste antimatter targetting small groups (3 or 4) of enemy units.  Better to be overly conservative than too liberal with the application and not have enough antimatter left over for when it counts.