Question 1: If I complete all of the conqueror requirements, but skip some of the slayer and destroyer requirements, do I still get the stash tab? Cause I freaking hate set dungeons so I am getting hung up on that requirement, but can easily do all of the conqueror stuff.

Question 2: When is the stash tab awarded? Immediately upon completing the conqueror requirements, or not until the season ends?

Question 3, and I think I already know the answer.... I only need to complete conqueror on soft core OR hardcore, and it will give me an extra stash tab in both cores, right?

You have to complete all the objectives, up to Conqueror, to earn the seasonal stash tab reward.
As soon as you complete all your objectives your stash tab will automatically show up.

I recently started playing Diablo 3 again after being away from it since 2015.

So I only have the stash tab rewards from Season 13 and 14 (this season).
You can earn up to four of them total. So I have two of the four at the moment...

I'll be able to earn another stash tab reward in Season 15 (starts in late September) and 16 (starts in early January).