You should pick up the two professions in Stormwind (assuming you are Alliance) and when you look in under professions you’ll see that you have a separate bar for the different expansions.
With tailoring the mobs in e.g. Classic zones still drop cloth according to the “old” ranking of zones, so you get runecloth from plaguelands, wool from duskwood, and so on. Enchanting I assume you would either disenchant items picked up in those zones or DE the cloth items you make in Tailoring to level it.
With the large volumes of cloth and low prices on the AH from people running old content you could save a lot of time just buying the cloth you need.
Yes, you can skip any of the different tiers (for want of a better word). So you might want to just pick up the professions in Stormwind, then if you want to level Outland tailoring (for the carpet e.g.) just pick that up in Shattrath, or pick up Northrend tailoring in Borean or w/e (think it’s Valliance Keep but check Wowhead).
So, arguably, unless you are a completionist, or want specific recipes, you could just pick up the profs to start with then ignore them until you hit the BfA zones. It’s up to you.
Yes, as per above, you can do one or the other, both, or neither. If you do decide to do e.g. Outland Tailoring, see the trainer in shattrath as said above.
you can pick up Kul tiran crafting as soon you go to boralus, starting from 1/175. So learning old profs levels is pointless for most proffesions as the crafting has been split into expansion tiers, so you can start right of the bat with crafting in kul tiras, not needing to complete all the lower expansion crafting anymore.