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.
What level find familiar is cast at depends on how the warlock casts it.
When she gains the Pact of the Chain class feature, the warlock learns find familiar. It doesn't count against her spells known, but nothing in the ability text prevents her from casting it with her spell slots. If she casts it with her spell slots, it works exactly like any other spell she knows how to cast. If she is 5th level when she casts it, she casts it as a third-level spell.
In addition, the warlock gains the ability to cast find familiar -- and only find familiar -- as a ritual. When a spell is cast as a ritual, it always uses the normal spell level of a spell. When the warlock casts find familiar as a ritual, she casts it as a first-level spell no matter what her warlock level may be.
Best Answer
Your options are found in two different places: find familiar and the Pact of the Chain feature description.
We begin with find familiar:
Normally, familiars sumoned by find familiar cannot attack. Pact of the Chain changes this:
So you can replace one of your attacks with an attack by the pseudodragon familiar, at the cost of the familiar's reaction.
Whether or not you should do this is the subject of some discussion on this Q&A: For a Pact of the Chain warlock, is choosing to have their familiar attack generally the worse option for the warlock's action?
Notably, clever use of the Help action is usually the best thing a familiar can do in combat: Can a familiar use a Help action to grant advantage on an attack to a near by ally?
Breaking this down, here's a bulleted list of how this works:
Certain Eldritch Invocations can improve the familiar.
Most notably, the invocation Investment of the Chain Master from Tasha's Cauldron of Everything significantly improves the familiar:
With this feature, when using your bonus action to command the familiar to attack, it is somewhat unclear exactly when the familiar takes the attack action, see this Q&A where I argue that it waits for its turn before taking the attack action: When does a warlock's familiar make their attack if the warlock has the Investment of the Chain Master eldritch invocation?
There are several more invocations that interact with the Pact of the Chain familiar, they can be found here on DNDBeyond.