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.
Channeling a spell through a familiar ends their invisibility
As you've already determined, the line about "...delivering the spell as if it had cast it" is enough to trigger invisibility ending on a spell cast.
Any of them would lose their invisibility after delivering a spell
Quasits and Imps can't normally cast spells so their invisibility blocks don't list "or casts a spell" as an end condition. Since all the invisibility stat blocks do list all of the offensive actions the monsters are normally capable of as ending invisibility it's pretty easy to figure out what should happen here. So all of these monsters will end their invisibility if you channel a spell through them.
A strict RAW ruling would disagree, but 5e just isn't written tightly enough to support that kind of ruling.
Best Answer
The Shapechanged imp is not actually a spider, so is not an eligible target for giant insect.
The imp’s Shapechanger trait says (emphasis mine):
The form only resembles a spider, it is not actually a spider. Further, its statistics other than speed are not changed, so it is definitely not actually a spider; it is an imp that looks like one.
Since the giant insect spell requires you to target an actual spider, the imp cannot be transformed using giant insect.