Agreed that there are too many powers and abilities here, and they're too far out of scope for a familiar. Let's see..

• blindsight: the only base familiar with blindsight is the bat, and his is echolocation. So, not inherently broken, but rare.

• burrow speed: Only familiar with a burrow speed, and tries to make up for it by tanking speed in general. This one is also... a bit odd. Why would an eldritch crystal have a burrow speed? More to the point, it lets you get in and out of certain places rather more easily than you'd otherwise be able to at low level, and gives it a near-guaranteed escape if it's on a surface it can tunnel in. Would be acceptable as a Unique And Shiny Thing, or convert it into a flight speed (for levitation). Possibly remove the ground speed entirely and give it only a flight speed (probably a slightly faster one) to represent the fact that it's a crystal and does not actually have legs. Lets it be a bit weirder (I suspect the player likes weird), but not in a way you need to worry about.

• otherworldly resistance: This is... excessive. Its primary effect is that it lets the player throw around area effects a much as they like without hurting their familiar. It's also not well-justified. In fact, as far as I'm aware, nothing in the game has blanket immunity to spell damage. If he were building a completely different familiar, I might accept immunity to a single element as the one completely awesome thing it had, but that's not what he's building here. Carve it out.

• innate spellcasting: Blur is second level, and increases survivability in a way that other familiars can't match. Meld into Stone is third, and offers both survivability and shenanigans. Both are overpowered, compared to what is expected of familiars. The only other familiar with a spellcasting effect is the octopus, who's water-bound.

• intelligence stat: 10 is way too high for a familiar base. all other base familiars have between 1 and 3.

• constitution stat: higher than any of the base familiars in the book (most of whom have 8). By extension, this is the only familiar with 3 hp. Probably pretty harmless in itself, but underlines the min-maxxing.

In general, then, the familiar as presented has far more survivability than any of the base familiars, as well as being weirdly intelligent.

Suggestion: As a base, keep the blindsight. Convert the burrow and land speeds into a moderate fly speed (levitate). Let him have the con. Drag the base int down to the 1-3 range (possibly with a special feature to let him use the arcana skill with a non-negative modifier). Ditch the otherworldly resistance and the innate spellcasting. At that point, you're a few notches down from a bat (significantly slower flight speed, no land speed, and no attack. The Arcana thing is cute, but not a big deal, and the other advantages are relatively meaningless). You can probably fit in one more interesting thing without breaking the bank. Possible "needs not eat/sleep/breathe" and make his slowish (10 or 15) fly speed also be a swim speed. That lets him explore (slowly) into underwater areas without having to be aquatic, and lets his master cast spells based on noxious fumes at areas that contain him without concern. In the end, it wouldn't be doing anything that other familiars couldn't do, but the combination would be rare and useful, and it would be doing them in a different way, which should appeal to the player involved.

edit: your player has expressed a strong desire for fire immunity. That's a specialized but situationally very powerful effect, which his own tendencies are going to make somewhat less situational. If you want that, it should be pretty costly. Given the character in question, I'd suggest you lose the "needs not eat/breathe/sleep" and swim, lose the blindsight, make the familiar a permanent light source and possibly give it a familiar-esque 1 damage melee attack (fire). Call it a Forge Spark - a sort of least fire elemental associated with the character's god.

The implications are minor, but not totally zero. To address your specific questions:

1. The type of spirit does not cause any visible differences, unless the player or the DM want it to. That would be a flavor/reskinning call, not a rules or mechanics call. Note that the spirit type and the shape taken are not linked or restricted in any way. A stark example is if you are a Pact of the Chain Warlock, you familiar can take the shape of an Imp, Quasit, Sprite, or Pseudodragon, and this is given no limitation as to the type of spirit. So you could have a Celestial Quasit.

2. The type may have an effect on the spirit's alignment, but again, that's up to the player and the DM. Again, that would be a flavor/reskinning call, not a rules or mechanics call. It obeys your wishes and commands regardless. It would be up to him to decide if his Celestial Quasit is mannerly, polite, and never uses profanity.

3. Yes, the default is that the character gets the same spirit back, just in a new shape. The caster does have the ability to dismiss a familiar forever, so presumably if he does so, casting it again will get him a different spirit, and could choose a different type. Again, flavor/reskinning issues, up to the player and the DM. Certainly, if the DM allows him to change the spirit, getting one of a different type would mean a new spirit, and thus a new NPC, with no memories of what the last one experienced.

The main actual effects, as far as the rules go, are that the type of the creature is celestial, fey, or fiend, (and not beast), and this may affect what spells it is affected by (e.g. immune to Animal Friendship, but may be subject to Magic Circle. Also, depending on its type, a Paladin's Divine Sense may detect it, and that could tell them something about the character (or at least who he associates with).

The point of it being a spirit, from a mechanics standpoint, is mostly to explain where it comes from, why it is obedient and maybe more intelligent than normal, and to provide a mechanism for summoning, re-summoning, and dismissal.

How much control the character has over the form of the familiar is, again, up to the player and DM. If I, Gimble the Great, decide I want an owl, is that all I can specify, and the spirit decides if it will be a 4-ounce burrowing owl, or a 5-pound snowy owl, and all the details of its appearance? Or could I specify the species of owl, but it gets to decide the coloration? Or do I get to pick all the details myself?