There is no real distinction to be made between extradimensional spaces and demiplanes — they're just different words for the same thing. Per the Dungeon Master's Guide, page 68, “demiplane” is just what extradimensional spaces of notable size are conventionally called. You'll also note that DMG page 99 equivocates “demiplane” and “pocket dimension”, saying that either is large enough to serve as the exotic location for a dungeon, so there isn't a real line that can be drawn between those by size, either.
So both “demiplanes” and “pocket dimensions” are extradimensional spaces, and the difference between the two is more a matter of language being vague yet evocative: the two different words just lend nuances to describing the same thing, like how “boulder” and “rock” both mean a chunk of stone, just one suggesting greater size. The difference in the words “demiplane” and “pocket dimension” is just suggesting (vaguely) something about the size of an extradimensional space.
So given that one is just a synonym for the other — yes, a familiar's pocket dimension is an extradimensional space, and could be also called a (very small) demiplane if you wanted to.
No, the extradimensional space is not attached to the wizard in any sense. If it was, there would be very clear wording about the danger of the wizard doing things like entering portable holes or rope trick spaces. That's not something that would be accidentally left out of the description of find familiar. Given that lack, it's safe for both master and familiar to enter extradimensional spaces like demiplanes and pocket dimensions. Unlike a bag of holding, there is no portal attached to the familiar (nor is there rules text about the danger) — think of it instead as the familiar being “tuned” to that personal pocket dimension, so the magic of the spell can always “find” its home pocket dimension.
The spirit is up to you, the mechanics are generally determined by the spell.
As a DM, you can absolutely work with your player to utilize a spirit that fits with their design. You can always go much further than that, but in general, the mechanics of find familiar are clear.
Pocket Dimensions are separate
This question covers more about pocket dimensions and what can/can't go into them and what that pocket dimension entails. You always have the right as DM to expand on that, but if you feel that by doing so it creates problems with your world or gives a character more capabilities than they should have, then you can stick with the rules as written and keep it limited.
The takeaway: I included this to specifically address the question on bringing something back to the pocket dimension. You can't unless you as the DM allow it. Find Familiar functionally would just call this fragment of the god, which resides in it's own pocket dimension, and send it and it alone back when dismissed or killed.
Best Answer
So, to your first question: The pocket dimension will be unique to your familiar.
While it's difficult to prove this, a dimension that potentially contains the familiars of every spellcaster with the Find Familiar spell in the entire universe at the same time really doesn't sound like a "pocket" dimension.
Also, what would happen if a Wizard sent his familiar to the pocket dimension at the same time as the vastly more powerful familiar of his Warlock nemesis? You have to assume that the Wizard probably wouldn't ever see his familiar again, if you see what I mean.
To your second question: RAW, the familiar probably can't take items with it.
The spell says you dismiss your familiar, not your familiar and everything they're carrying. Similarly, "it" disappears into a pocket dimension - "it" here definitely refers to your familiar, there's no reason why it would mean "your familiar and everything it is holding". It's also worth mentioning that your familiar is defined as a spirit that takes the form of an animal - so maybe it could take spirit objects with it, but probably not physical ones.
Additionally, when asked a similar question for Find Steed, Crawford said that the steed should leave gear behind. While Find Steed is different to Find Familiar, its wording with respect to this issue is similar, so the same rules probably apply.
On the other hand, Mearls says he'd allow it, but only for a few minutes. The usual caveat applies here, of course: Mearls is not a source of official rules interpretation, just a source of how he would rule things in his game. So you can pretty much take or leave his advice as you choose.