Resource Management during the Adventuring Day
As a class, the Paladin isn't overpowered. The Paladin does have a nice mix of weapon and spell skills to offer any party.
Discussion
On page 84 of the DMG the adventuring day around which the game is balanced calls for 6-8 encounters of medium to hard difficulty, and 2 short rests. If your DM isn't running that much challenge, then the Paladin's resource management problem isn't being put to the test as the game balance intended it to.
A Paladin can do large bursts of damage, for sure. What he can't do is burst damage all day, every round, for that many encounters.
Melee Attacks Available
You mention that the Paladin is now 5th level. This is a threshold in melee damage increase thanks to having gained the second attack per melee round. The initial increase in apparent power should level off a bit as you increase levels and the DM increases the CR of the monsters you face.
He has almost died at times
If he's playing "high risk - high reward" style, over time the odds will catch up with him. It appears that the shield spell that goes with his armor is giving more chances to not get hit. He is thus able to play a higher risk combat style than he might otherwise. If it allows him to do two handed weapon attacks, and still benefit from a shield through magic, that's more powerful than a paladin who has to use a shield and a one handed weapon to handle the rigors of melee combat.
That combined with the great weapon master feat might be what's causing this perception of great power. It might also be the pace of your adventuring day.
Spells Available
Per Long Rest, at 5th level, he has 4 spell slots of 1st level spells and 2 of second.
If he doesn't have to concern himself with casting any other spells (this has to do with your party make up and who the other spell casters are) then the paladin can use divine smite six times per long rest. That's a nice pile of bursty damage during some fights.
Is that "overpowered?" If you only have a few encounters per day, it can look that way. This is an adventure design challenge for the DM to address, as you both suspect.
Encounter design is explained on pages 81 - 85 of the DMG. The proposed XP budget for encounters and the adventuring day is built around a party of 4 characters, and the challenge ratings that accompany that XP budget. (For a party of 5th level adventurers, that's 3500 XP per character per adventuring day, 14,000 XP worth of monsters). If the DM reviews that and compares it to the challenge he's been offering you, he may choose to ramp up the difficulty.
A character can only have one long rest in a 24 hour period [PHB, p186]. This serves to limit the power of abilities and spells.
FWIW: in our party, the Paladin is often either the hero or the goat. He who lives by the burst, dies by the burst. He has a +1 magic axe (just found, has not used in battle yet) and some +1 bit of dwarf light plate. He does not overpower the rest of the party but he can hold his own.
In a tweet Jeremy Crawford (lead rules designer) advised:
Q: Is divine smite supposed to be limited to once per turn? Can I use it three times with extra attack and DW?
A: The limit on Divine Smite is the number of spell slots you have available to spend on it.
I think these seems like a lot of fun, and I'd probably allow it in my game as well. I've created custom familiars in the past, and most recently I've used this "5e Familiar Handbook" I found. If you're not sure about your custom familiar, compare it to the others on this list. I like how you kept the climb and fly speeds low, but I think I'd personally do something like this: Speed: 20 ft.; Climb: 40 ft.; Fly: 20 ft. With those movement speeds, you'd have a clumsy-lookin' flyer, but an excellent climber, much like what I imagine a flying monkey would move like.
Best Answer
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.