Running an M+ dungeon is a roughly 30 minute ordeal (usually less) + however long it takes you to find a group and get to the dungeon.
If you have contacts, that bit is probably ~10 minutes tops.
If you’re pugging as a tank, healer, or DPS with a good IO score, it’s probably not going to be too much more than that either.
If you’re DPS and your score is meh, you might be stuck applying for awhile. COULD be 10 minutes, or it could be 45. Not likely to be “hours and hours and the entire week” like some people try to claim though.
Anyway, Mythic+ is pretty accessible to people with limited time and varied time. You don’t need to be available at the same time every day/week to be able to succeed in Mythic+. The time investment is relatively small and the player count is significantly smaller than raiding, so it’s easier to set up impromptu runs without worrying about a set schedule. You should be fine in that department.
As far as raiding is concerned, you’d have little problem finding a guild that will take you to Normal raids (assuming you find one that cares to bother with that content when benthic gear exists). You MIGHT find a guild that has a core group push through Heroic and doesn’t mind dragging a couple extras through once it’s farm content.
But you’re going to have trouble finding a group that will take you as a core member for progression days in Heroic and definitely Mythic. It’s one thing to miss a day here or there because something came up, it’s another to miss every other raid day if not more because your schedule is unpredictable.
It’s not impossible to find a group though. We ran with people who could make one day and not our other day. We ran with people on volatile Retail schedules. We ran with people who had to take temporary hiatuses. It’s fine to some extent. The consistent but not quite a fit stuff was easiest to work around. Just had to recruit the right people to fit around it. The straight up volatile schedule stuff is worse. 1 or 2 people is… manageable. If it’s someone in a DPS role and you don’t depend on them to partial-carry lower performers.
But we raided Heroic. It’s going to be even harder if you want to push Mythic, because filling your slot if you don’t show up is much harder since you generally won’t have the ability to pull from cross-realm.
If your schedule is volatile, probably the best thing to do would be to find a late night / early morning guild that avoids your schedule entirely due to the time, rather than days. But I don’t know exactly what you’re dealing with, and that’s almost guaranteed to cut down on the amount of sleep you’re getting. I can get by with 5-6 hours on the regular just fine. I know people who are totally fine on 4. I know people who can’t function without their full 8 or more. So I get that this may not be an option for you. Doubly so if you’ve got family to worry about too. Easier to justify a few hours for raids if you’re not working that day and will have time to spend with them too.