I picked up a dwemer axe of absorption, but sold it before I got the letter from Calcemo. If I were to find another axe and enchant it myself, would it be acceptable & the quest can be completed?

It can be any. Infact it doesn't even have to be enchanted.

He actually asks for things besides ingots? I keep getting letters when I buy an ingot, where he desperately wants it or another. It's crazy. I could give him a 100 if he'd leave me alone about it.

It does have to be enchanted if he asked for an enchanted one. In all of my games he always asks for Dwarven Bracers of Major Alchemy, whenever I get them. It wasn't until recently that I realized he could ask for anything else.

But, to the OP's question, I don't know if self-enchanted ones will work or not. Usually I just wait until I find another one of whatever he wanted, since his letters always tend to arrive a day or so after I've sold whatever it was he wanted.

Yeah, Calcemo wanted a dwarven bow I'd just bought and upgraded and enchanted for myself. I went back to the vendor a couple of times until I could buy a new one and took it to him. He asks for precisely the weapon you received or found in a dungeon, so with or without the existing enchant depending on what you find or buy. You'd have to sacrifice the one you found or have the enchant to put on it. He paid pretty well.

In my game, he just showed up as soon as I picked up a dwarven sword for the first time. I just gave him a replica as I sold the one I already had. I don't think it has to be exact one. He asks for a dwarven artifact in the note and nothing specific.

He asked me for some enchanted Dwemer Cuirass I found in some ruin. I didn't get any further requisitions from him after that.