Yes, Pals can carry and transport items in Palworld. They primarily function to transport resources (wood, stone, ore) from the ground to chests within your base if they have the Transporting work suitability. Additionally, having specific Pals with passive skills like "Cat Helper" in your party increases your personal maximum carry capacity.
Pals will always transport to the nearest chest that is allowed to hold the item and has space. There are a few exceptions with special chests such as refrigerators, guild chest, or pal food boxes. In general the only way to control it is to disallow that product type in every box except the one you want it to go into.
Transporting Pals are creatures that are able to move items from storage spots to gathering and mining stations. For example, Transporting Pals can bring Stone from a Stone Pit to a Chest.
Transporting is one of the Work Suitability skills a Pal can have. It allows Pals to move items from gathering sources and ground drops to appropriate storage units within a base.
Cattiva: This Pal provides a +50 carrying capacity! Lunaris: Carrying capacity increases by +80 when you have Lunaris in your party! Broncherry: Both regular Broncherry and Broncherry Aqua contribute +100 carrying capacity! Kingpaca: Like Broncherry, Kingpaca also grants +100 carrying capacity!
You need at least half the Pals for transporting, if you want it to be efficient (or a couple Knocklem, possibly condensed). But you shouldn't need that. Stacking up to 9999 in each mining site takes time and you can easily just empty them yourself whenever you come back from an adventure.
Whenever you pet your Pals, their Sanity meter will go up by exactly one point. That's not a lot, and there are much better ways to increase a Pal's Sanity so that they stop slacking off, but those points can add up quite a bit over time.
Yes, they do come inside, but for me, it was a mixed bag. Some nights you could see them march inside the building and get in their beds. Other nights some would get in, and the others would sleep outside the door like dummies.
No, CX404 from the Fallout TV series is not the exact same Dogmeat as in Fallout 4, but she's the show's version of the iconic companion, a female Belgian Malinois connected to the Enclave, while the Fallout 4 Dogmeat is a German Shepherd from the Commonwealth, though both share the "Dogmeat" name as a nod to the recurring canine in the franchise. CX404's backstory (Canine Experiment 404) ties into the Enclave's breeding programs, similar to theories about the Fallout 4 Dogmeat's origins, making her the show's unique take on the classic character.