I can't get Neloth's quest, Telvanni Research (the one involving the harvesting of Spawn Ash), to appear for me. I've done all of the other quests in Tel Mithryn, aside from Healing a House (that one won't appear, either; sometimes, when I talk to Neloth, he says something about one of the towers being damaged, but I have no dialogue option to bring it up to him). The quests Azra's Staffs and Heart Stones are the only ones that appear, and they seem to be repeatable. Is there anything I'm doing wrong, and anything I can do to fix it?

You get the house one from his potion chick, she's in charge of fixing it. Iono about the research one.

Thanks for the Healing the House one... Can anyone help with the Telvanni Research than this?

Really? No one can help?

It seems to turn up randomly.
On my last playthrough I got it early but this playthrough I only got it as one of the last quests.
I had already done everything else and was seeing Neloth for a Black Book quest when the dialogue option to help him with his research popped up. I finished the quest for him as we were walking to the Dwemer ruin.

I suppose youll have to check in with him every couple of days to see if he has a new quest, I noticed they do not appear right away when you turn the old one in but after a few days.
Good luck