It's almost correct, but not quite.
There are two separate, unrelated ways to get a familiar in D&D 5e, and you're trying to combine them.
The first method is by use of the Find Familiar spell, which allows you to summon a celestial, fey, or fiendish spirit that takes the form of any of a list of creatures. This list is expanded by the warlock pact of the chain. This familiar is perfectly obedient, can be resummoned when it dies, can be hid in a pocket dimension, deliver touch spells that you cast, and everything else specified in the spell description. This method gives you a familiar with the basic stats of the chosen creature, not the "variant: familiar" traits of the chosen creature (unless your DM chooses to have that creature type appear).
The second method is by finding a quasit, imp, or pseudodragon that has the "variant: familiar" trait (which is 100% up to the DM), and enlisting it as a familiar by interacting with it. This familiar has only the traits listed in the stat block for that creature, including the variant traits, but none of the traits of familiars given by the Find Familiar spell. No pocket dimension, no touch spells, no limitations on what actions it can perform, and if it dies, it's just dead. This type of familiar is an NPC controlled by the DM (much like a hireling or other follower), and is only as obedient to the PC as the DM says it is, using the MM entry as a guide.
Now that we've established how things actually work, we can address your real concern. Your warlock can't yet communicate at a great distance, but he can soon. There is a warlock invocation available to him called Voice of the Chain Master that does the same thing, but with unlimited range on the same plane. As you've realized, this ability has some incredible potential, especially for scouting.
It's not any more powerful than other options and should not be limited.
Let's compare it to some other invocations. There's one that lets a warlock cast Disguise Self as at will. This would let him see a guard, and appear exactly like that guard and just walk around the enemy camp unimpeded. Or maybe impersonate the leader of the camp and just take it over without even a struggle. There's another invocation that lets the warlock cast arcane eye at will, which gives you a way better scout than an easily killed creature. A familiar, even an invisible one, still has to succeed on a Dexterity (stealth) check to avoid being heard and then easily killed. An arcane eye does not.
So, in order to fully utilize this scouting ability, your warlock has to pick the chain pact and spend one of his few, precious invocations, both of which are huge opportunity costs. He deserves something in return. This something is you not limiting it. It already has a flaw in still being able to be killed by anything that hears it, or smells it. That's right, just about any pet wolf is going to be enough to catch this familiar. It doesn't need any more limitations.
Yes and yes.
It is all in the wording of the spell and feature.
it is a celestial, fiend or fey (your choice)
Implies that you have a familiar and pick what it is, regardless of its form.
If you cast this spell while you have a familiar, you instead cause it to adopt a new form.
Implies your familiar just changes form, but remains being what it is (e.g., Celestial).
choose one of the following special forms
Implies you have access to new forms.
So, yes, the familiar can take the imp shape, and it is still a celestial (assuming your previous familiar was one.
The creature types are Fey, Celestial and Fiend
For a simple summoning, Sprites are fey, while Quasits and Imps are fiends. That leaves the question open for the pseudodragon. It can be a fey, celestial, or fiend in the form of a pseudodragon (as the spell description states). Nothing in the pact description contradicts that.
Without getting complicated, three out of four special forms already coincide with fey, fiend, or celestial. The pseudodragon (form) can be any of the three, since the spell's features otherwise hold in this case. (PHB p. 107).
The form and the spirit don't have to match if the Warlock wants to mix it up. That could be fun.
Mix and match? Sure, nothing prohibits it
The familiar is a spirit that takes a form. That's as specific as the spell gets, beyond it having to be a beast (for a regular casting of find familiar - beasts from the list), or the additional choice of special creatures if it's a summoning from a Warlock with Pack of the Chain. (The Warlock could summon a weasel form if they wanted to; they don't have to summon a special).
If the Warlock so desires, they can summoning a fiend spirit in the form of a sprite or a celestial spirit in the form of an imp. All that changes is the form; that's a special feature of the Chain Pact. Whether or not that makes sense is up to the Warlock in question.