The Find Familiar spell is currently the only method for a wizard to obtain a familiar, and it lists the creatures available. However, a large number of creatures in the Monster Manual, such as the pseudodragon, imp, or quasit, include the 'Familiar' variant. Note that a Warlock with the Pact of the Chain feature can obtain many of these creatures as a familiar.
What you have to remember is that variant monsters, like monsters, are designed for the GM to use to make enemies more interesting. The Familiar variant is a monster variant, just like the troll's Loathsome Limbs variant or the Genie Powers variant. They're for GM use rather than player use.
The Mage NPC in Appendix B of the Monster Manual also has a Familiar variant, which says:
Any spellcaster that can cast the find familiar spell (such as an archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell, (see the Player's Handbook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit.
So the Familiar variant is for GMs to create more interesting NPCs, rather than to provide players with additional options. Of course, with your GM's permission, you could obtain a more interesting familiar. This would probably involve actually finding such a creature and somehow forming a bond with it. But this relies solely on your GM to allow and arbitrate.
It's also worth considering that allowing a wizard to obtain one of these more powerful creatures somewhat invalidates the Warlock's Pact of the Chain.
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.
Best Answer
RAW, No, but...
As you've said, find familiar creates a
However, talk to your DM.
This would definitely fall under the realm of things I as a DM would allow, especially because it fits in the roleplay department and doesn't change balance. If you talk to your DM, I bet they'd ignore RAW in this specific instance.