5th edition was designed to not require magical items. A party of level 20 characters can win a fight against a CR 20 dragon with no magical items whatsoever.
That said, one very important aspect of magical weapons is that they bypass resistances and immunities of creatures. Without any magical weapons, a fighter is almost useless against a creature immune to physical damage, unless he has any magical weapon. If you don't give any magical weapons, you can just not use monsters with immunities to avoid the issue.
In conclusion, 5e works just fine with no magical items, so long as the DM is mindful of resistances and immunities of monsters.
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.
Best Answer
RAW, the minimum CR is 1
Dealing with CR is always at least a little bit iffy, and it's more a math game than a way of determining how special or interesting a creature is. That is to say, it's less that a creature gets Legendary Resistance because it's interesting but more that it's interesting because it has Legendary Resistance. More subjective thoughts on whether or not a creature is cool or interesting do not factor into CR.
The DMG has guidance on how Legendary Resistance affects CR
For determining whether or not a homebrewed creature can or should have this feature, the DMG suggests considering Legendary Resistance to add effective HP to a creature, based on its CR without the feature, for each per-day use:
So you can give this to any creature you like, but it will potentially alter the CR to be greater than you may prefer. This prescribed method allows you to estimate that impact (to the degree you can ever reliably determine CR).
That the guidance has a value for CR 1 creatures definitively demonstrates that the minimum CR the rules define as valid for attaching Legendary Resistance to is 1. But can a creature that actually has Legendary Resistance available three times per day have a CR of 1?
An example by the numbers
Let's look at a pretty straightforward example by modifying a creature that is already CR 1 to include Legendary Resistance (3 per day) to see how the CR changes. We'll use a CR 1 creature because the guidance does not have any suggestions for challenge ratings below 1. We will also use the "Modifying a Monster" guidelines given in the DMG (Chapter 9: Dungeon Master's Workshop, Creating a Monster, Modifying a Monster), especially the "Monster Statistics by Challenge Rating" table.
A Direwolf is a CR 1 creature as written:
Defensive Rating
Offensive Rating
It deals an average of 10 damage per round, which corresponds on the table to a suggested attack bonus of +3
The Direwolf also has the Pack Tactics feature, which the Monster Features table suggests should raise its effective attack bonus by 1
The Direwolf's actual attack bonus is +5, which is only 1 point greater than the effective +4 indicated by the previous two bullets and so the Offensive Rating does not need further modification
The Offensive Rating is 1
Average Challenge Rating
The next step is to take the average of the Defensive Rating and Offensive Rating, which is ((1/4) + 1) / 2 = 0.625. We are then instructed to round up to the nearest CR, which in this case is 1. So the listed rating of CR 1 checks out according to the "Modify a Monster" rules.
A Direwolf with three uses of Legendary Resistance per day
Now let's add three uses of Legendary Resistance per day. The guidance suggests that this should count as an additional 30 effective hit points for a CR 1 creature, giving us an effective total of 67 hit points.
This moves us one row further down the Monster Statistics by Challenge Rating table, from a 1/4 Defensive Rating to 1/2, but that row still has a suggested AC of 13.
None of its other statistics have changed, and so the formula for Average Challenge Rating is (0.5 + 1) / 2 = 0.75. This also rounds up to the next CR, and the next CR is 1, just as before.
By following the "Modify a Monster" rules, we start with a CR 1 creature (the minimum for which the DMG offers guidance), give it three uses of Legendary Resistance per day, recalculate its CR, and end up with a creature that still has a CR of 1. Therefore the minimum CR for a creature which has that feature is 1.