which archtypes and cards are best against the deck?
As a Stax player, some of the more commonly maindecked cards I don't like to see include the following(in no particular order):
Force of Will(I don't always get a Welder)
Duress(see above)
Thoughtseize
Island(or basic lands in general)
Mana Drain
Rebuild
Merchant Scroll-->Rebuild
Aether Vial
Goblin Lackey
Juggernaut
Kataki
SB cards like Energy Flux and Art Mutation wreck Stax as well. Though I hate to admit, Leyline of the Void is also usefull vs. Stax. But keep that on the down-low, we don't want that information to spread.
Pre-restriction, Gifts Ungiven was a pain in the arse to play against. But Merchant Scrolls still allow blue based decks to run a singleton Rebuild.
Oath is only bad if they get it out soon enough. A turn 4 Oath isn't too scary to me. Sometimes I'm able to amp up my Smokestack counters just with Orchard Tokens.
As for Goblins, I wouldn't go the Food Chain route. A R/G Goblin deck with 4 Lackeys, 4 Aether Vials, AND 4 Tin Streets would work though.
Shop Aggro is also a pain since they typically run welders as well.
I tested quite a bit vs. that R/G list that won first place, and as a Stax player I wasn't all that impressed. I can't even say that the matchup was better than 50/50(for them) from my experience. Chalice @ 1 wins the game if they don't see Tin Street Hooligan soon. Not to mention that Vial'ing in a TSG does nothing.
Null Rod vs Stax?...not so much imo because some Stax decks still run the Rod. Mox Monkey in a deck with Welders is SOLID vs. Stax. In fact, Mox Monkeys are what I miss the most when playing against the mirror or another Shop deck. Once you kill that first Mox, you can flip and eat all their artifacts.
Honestly if your meta is predominant with Stax, a mostly blue deck with FoW, Drains, Merchant Scrolls, and Back to Basics is an option. Especially with Energy Flux in your SB. Stax can't really deal with enchantments.
Mike