Yes, you can add a stone to an enchanted weapon, they coincide and do not conflict. As far as what to enchant, there's 3 main stones I like: Rubies, Topazes, and Aquamarines. Jades in practice add way too little to be worth it. Emeralds are kinda ok but in most cases (all but when you have a monster pushed against a wall) weapon speed doesn't matter that much. Amethysts typically actively work against you and your ability to kill things.
Rubies you would use if you were using Fighter and Brute for your combat perks, that combo results in the highest minimum damage, and the most consistent kill speeds.
Aquamarines you would use with the Scout and Desperado combat perks, for the biggest possible crits, and highest possible average damage.
Topazes you would use ideally on top of a weapon with a built in defense bonus, for when you're planning on ignoring enemies.
In my current active save I've got a Galaxy Sword with 3 Aquamarines in it and the Crusader enchant, which is my combat blade. I use that when I want to kill things. I plan on replacing it with a Dragontooth Cutlass once one is found, or upgrading it to the Infinity Blade when possible, whichever comes first. I have a Dwarf Dagger (it was a Dwarf Sword until very recently, finally got the dagger to drop!) with 3 Topazes in it (no enchant) for when I'm just ignoring enemies and blasting my way through, such as during Skull Cavern Invasion missions. And I have a Lava Katana with 3 Aquamarines and Haymaker on it, for when I'm in need of Fiber or Hay.
Personally I've tested both and Wildly preferred the crit build over the flat damage bonus build, in practice it just feels so much better. Most of the time I'm just swiping at enemies coming at me and continuing to run, and with the crit build, sometimes that one hit just kills them, and by sometimes I mean like 1/4 times. If it doesn't, I keep running, and if they come back, rinse and repeat. Plus the crits feel So nice, and the speed with which you mow through enemies is impressive. Nothing wrong with the Ruby build of course it just wasn't for me. Biggest advantage of the Ruby build imho is if you are running Fighter / Brute, and have a Dwarf Sword for defense with Topazes, it still has a 25% damage boost from the perks. A Dwarf Sword or Dagger with Topazes while running the crit build is almost entirely useless, your huge crits just don't matter because you don't proc them nearly enough. That's fine with me personally but definitely noteworthy.
I did build a combat calculator if you want to play with some combos and see how a given setup stacks up against some enemies, if you feel like checking it out:
docs.google.com
Oh and a final note, I'd recommend changing the thread title to contain a (spoiler) warning, maybe even make the title a bit more oblique, just because this all is late enough of content to be considered spoilers.
Edit: For Tool Enchants I have the same as Hill Myna except Bottomless on the Watering Can (in no small part because it looks cool, and also because I just can't be bothered to refill it) and Generous on the Hoe, for extra Clay and such. Also my take on Emeralds was based on Swords, I can see how one would help a Hammer out for sure. I use literally the same rings for "easy" content, and wear an iridium band / crabshell ring and lucky ring / crabshell ring for more "dangerous" content, as well as for the Volcano if I'm just running through to check the treasure chests.
