Be conscious that gathering professions MAKE gold, while crafting professions COST gold, to buy the materials they need if you are not gathering them yourself.
Crafting professions cost quite a bit in materials just to get your skill up to learning all the recipes. After that, you might think that you would turn a profit buying materials and selling finished goods, but it is often not so. Long story.
Engineering used to have a lot of gimmicks you could use in PvP, but most of those were removed in the cause of balance back in Legion.
Another thing you won’t understand until you play is that gathering professions depend on being out in the world a lot, whereas making gold from crafting professions are all about buying, selling, addons, even spreadsheets, and spending time at the Auction House. You won’t be doing anything profession-related while you are doing PvP in Arena or Battleground instances, and Mythic+ Dungeons in instances.
Another thing you should be aware of is that it is really not worth taking up professions until you enter Shadowlands at level 50. All demand for everything below 50 has been suppressed, and you can learn Shadowlands Professions without having learned them in lower level expansions.
And also, in Shadowlands, if you like being outdoors, Fishing actually brings in decent gold as well.
Of the pairs you list, I don’t see Engineering + Enchanting as viable. Not only is Eng probably not worth it, but while Ench will bring in some gold from disenchanting gear you get, it won’t be enough to do much with Eng.
The other three are certainly possible. Like Minacara, I personally would go Herb/Alch. Herbs bring in the best gold of gathering professions, and both you and everyone else needs potions and flasks.
There is something to be said for Skinning/LW, as a rogue, but there is not a great deal of gear you can craft for yourself, and if you really want it, you can buy it aty the Auction House with the gold you make from gathering,