I'm level 41 now but haven't bothered doing the main quest in Morrowind (never gotten Corprus). Can I - without being the Nerevarine - do the various Tribunal quests other than chasing after the Dark Brotherhood?

I remember reading something on the Wiki that the different factions intend to use my Nerevarine status to their advantage.

The only time I have actually finished the Tribunal MQ I wasn't Nerevarine yet, although was pretty far in the Morrowind MQ. I could do everything, but at least Almalexia at one point spoils I'm Nerevarine. Tribunal thinks you've done the MQ in Morrowind but doesn't prevent you from doing any Tribunal's quests as far as I know...

I don't think it does. Until the end of the Tribunal Main Quest there is no indication that you are the Nerevarine. Almalexia realizes it to be true, but you do not have to have fulfilled the Nerevarine prophecies for her to realize this. She may have her own methods.

The developers seem to have thought of people trying Tribunal before the main Morrowind quest because nobody says anything about you being the Nerevarine except for Almalexia late on, and it's not like she's completely sane. Even if her sanity is unrelated she might just be saying you're the Nerevarine as part of her plan to kill you later in the Clockwork City. The random townspeople only act like you're the Nerevarine when you actually are, they tend to act surprised to see the Nerevarine in Mournhold.