Does sleeping increase capture rate?

Sleep sits in the same category as weakening or stunning a creature: all three push the success rate up before you throw anything. So the encounter is not really about damage, it is about what state the Aniimo is in at the moment the pod lands. A creature that is awake and healthy is the worst possible target, and there is rarely a reason to throw at one.

That turns capture into a setup problem. You have Anipods to deploy, traps to set, sneaking to close distance without being noticed, and the environment itself to work with. Sleep is one more lever in that kit, and it pairs naturally with the quiet approach, since getting into position without waking anything is half the job.

The practical habit is to stop treating the throw as the play. Get the Aniimo asleep, or stunned, or low, then throw. If you have none of those and the creature is already alert, you are spending pods on the worst odds the encounter offers.

When the setup stops mattering

Ultra Pods have a 100% guaranteed capture rate, including on Alpha Aniimo. Against one of those, sleep changes nothing, because there is nothing left to improve. Hold them for the targets you cannot afford to lose, and use sleep, stuns and traps for everything else.

Not settled: How large the sleep bonus is, and whether it stacks with a weakened health bar or simply overlaps with it, is not pinned down.