No, a Warlock with the Pact of the Chain feature does not receive Magic Resistance if they choose a Quasit, Imp, or Pseudodragon. In the first place, a variant rule is only in play if the DM chooses. However, even if the DM decides that pseudodragon familiars (for example) are a thing, it still doesn't benefit the Warlock.

Some pseudodragons are willing to serve spellcasters as a familiar. Such pseudodragons have the following trait.

So "some pseudodragons" will be familiars who share Magic Resistance with their masters. The Warlock, however, gets their familiar from the find familiar spell, which says:

You gain the service of a familiar, a spirit that takes an animal form that you choose: [...]

The Pact of the Chain says that:

When you cast the spell, you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.

In other words, the Warlock's familiar isn't any sort of pseudodragon, let alone one with the variant. It's a spirit that takes the form of a pseudodragon (and can take a different form anytime the Warlock chooses). The spell does say that:

[...] the familiar gains the statistics of the chosen form [...]

However, the statistics of a creature are defined on pages 6-11 of the Monster Manual. The short definition is that, if you look at a creature's entry, it's everything in the yellow box. This doesn't include variant traits like the Familiar trait - those are in green boxes off to the side.

All of that aside, a generous DM could, of course, allow a Pact of the Chain familiar to provide its master with Magic Resistance. It must be pointed out, though, that this is making the Pact of the Chain vastly more powerful than it normally is, so DMs thinking about it should consider carefully.

An even more generous DM could allow any player to gain the service of a creature with the familiar variant. This, too, should be considered carefully - it's effectively giving the player a more powerful version of the Pact of the Chain feature for free.

No

Creatures aren't things, they're creatures. It's pretty clear to me that invisibility's description (same as the spell in PHB) is meant to cover whatever equipment/knickknacks/clothing a creature is carrying around with it so that a pedant couldn't argue that you had to be naked for it to work.

Sadly by saying, "anything", we've got a RAW argument that when you get on an invisible horse, jump in the invisible orc's arms, or bizarrely buff up the imp and have it lift you that you should turn invisible too. By RAW this seems to work, but frankly 5e has lots of odd things that seem to work be RAW. You shouldn't expect this type of strict reading to be followed in most games.