If my familiar is holding something when it turns into back into a tattoo, what happens to the thing it's holding? Does it turn into a tattoo w/ the familiar?
Familiar Tattoo (Su): "Unlike most familiars, her familiar can transform itself into a tattoo that she carries in her flesh."
It doesn't say gear, just itself, so gear probably falls to the ground (or into a pouch, if the familiar rides around in one).
Related: Is there a way, by RAW, to turn a familiar into a pack animal? Say a compsognathus for instance.
Are you asking if a familiar can carry things? Why wouldn't it be able to?
Also, do I keep alertness when my familiar is in tattoo form?
Familiar Tattoo (Su): "In tattoo form it continues to grant its special familiar ability, but otherwise has no abilities and can take no actions except to transform from tattoo into creature."
I assume "special familiar ability" means the Special Ability it grants the master based on type. (+4 Init for your Compsognathus) However, the second table is also referred to as containing special abilities ("Special" in the table) based on level, not type. Since the Familiar Tattoo ability says "special familiar ability" singular, not plural, I think it means the former. So no alertness, touch spells, SR, speech, etc.
I think I mean not just carry but use pack saddles/backpacks, things like that. Like, can I give my compy a handy haversack?
There's not really any rules for that. Unequipped gear in Pathfinder is just sort of in an 'inventory'. You'll often find GMs who make up rules about how much stuff you can carry (regardless of weight) and how many quivers you can have and things like that. Most people, I think, don't find that fun and just ignore it.
I would say that anything that can wear armor can use a backpack, and anything that can wear barding can use saddlebags (or the equivalent).
While your tiny lizard has a strength of 8 and can carry an impressive 26 lbs without encumbrance, I would probably try to keep in mind it only occupies a space 2-1/2 feet across, so having it carry a bunch of greataxes might not work so well.
In the end, you'll just have to ask your GM.