Is there a particular setting that works best for autobattling - as in how many stars should go in each of bite, blow, and rush?

It depends on the highest attack of your pet and if they have any genes that would make using certain attacks more valuable. For example, if your pet has all normal genes with a Bite Damage of 24, Blow Damage of 23, and Rush Damage of 20, you probably would want to assign Bite the most stars, then Blow with the rest. 

But if your pet has a gene that makes certain attacks more useful, your best bet is to assign the most stars to the genes the attack benifits. For example, with a Unicorn has the Magic Horn gene. Magic Horn allows for a chance of your opponent missing the attack if you use Bite or Rush. So in that case, you would probably want to split the stars between Bite and Rush.

If your pet has an elemental attack, since those modify the Bite damage,  you should probably, in that case, assign Bite with the most stars. Same with other genes that greatly modify a Specific attack damage. For example, if you have a Roc with Spurs, since Blow Damage is the one modified, you might want to go with 5 stars Blow, 4 stars Rush. This is a good idea even in the case where Bite has a higher slightly higher attack than Rush, i.e Bite Damage at 31, Rush Damage at 29. This is so Blow can be the first move the opponent faces.

Taking into consideration how attacks affect other attacks aren't really useful for autobattling. This is usually most helpful with manual battling, as Rush can be extremely helpful in preventing those Unis with magic Horns from blinding you or pets with elemental attacks from attacking you.

So Bite prevents Blow.
Blow prevents Rush.
And Rush prevents Bite.
(This only applies if the move actually hits the opponent, so if you use Rush and the move doesn't hit the opponent, the opponent can still use Bite).

The most useful move when taking in consideration these rules is Rush, but you'd have to assign 4 stars to another move, which your pet would use before using Rush, so that's kinda moot. The only advice I'd give you using this rule is that if you have a pet in which Rush is the only move that is modified or has a gene benefitting it like Massive Wings for Rocs, or Ice Spikes for Lykos, maybe it would be better to assign 4 stars to Bite then the rest to Rush, even if Blow has a slightly higher attack than Bite (i.e Bite Damage at 21, Blow damage at 24). This is because as you can see, Bite prevents Blow, and most users use Bite. So if you assigned 4 stars to Blow, then the rest to rush, you can be easily taken out because a pet using Bite wouldn't even have to worry about you attacking back.

Wow, complicated.  Thanks so much for the overview!