Ok, testing:

Having 2 dwellers in a crafting room with -% crafting time does no good, the effect does not stack

+% Happiness apparently just adds to a dwellers % while the pet is equipped, and does not push it over 100%

anyone know anything about the +% health ones? health bar doesn't move when adding the pet to an injured dweller, so I guess it adds to their total possible health, and current health, by the %, also temporarily? or would it make sense to use them with level 1, 17 endurance dwellers sent into the wasteland so they get extremely high health at level 50, even without the pet?

The +% wasteland weapons and outfits, and the +% junk ones, I'm guessing would be useless, except maybe to reach an objective quicker? since every dweller goes until at capacity, and earlier loot would be common loot

+damage is easy to figure out, same with Damage resistance

x stranger chance, does it add to the chance for the room, or the whole vault?

I'm guessing objective completion pets also wouldn't stack, otherwise you could complete them real fast

+% XP seems pretty useless, a level 1 reaches 50 in one wasteland trip

Healing speed and rad healing speed also useless in normal circumstances?

Wasteland return speed is good, faster trips, though it only affects the shorter part of the trip

+wasteland caps might be good, if I wasn't at max caps

Where are people putting some of these pets? I'm guessing +damage ones in the first room, +objective completion as needed, wasteland return speed I'm putting on dwellers I send out with the +7 endurance gear, so I can get more dwellers in that gear faster