Are there specific rules that determine what happens if, let's say, my pet eagle lands on my shoulder in-combat?
Does it get an AC bonus? Do I move slower? Do we move together?
Or should I talk to my DM about making a new houserule?
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If your eagle lands on your shoulder, then so long as you are at least one size category larger than the eagle, the rules for mounts would come into play. To answer your questions directly, it wouldn't gain a bonus to AC, nor would you suffer a penalty to movement, but it would move with you on your turn without using its movement.
Note that the eagle would need to use half of its movement to mount you, RAW, although since it can simply fly onto your shoulder, it would be reasonable for a DM to ignore that in this case (I would).
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
Once during your move, you can mount a creature that is within 5 feet of you or dismount. Doing so costs an amount of movement equal to half your speed.
Being mounted by your eagle shouldn't impose any restrictions on you whatsoever, since you would be considered an independent mount:
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.