Do you ever get anything other than grey items?
It depends on the expansion. In Legion, mining mobs was productive.
Engineering some boss mobs in ulduar is still pretty profitable
I don’t think i’ve gotten anything other than greys from mining mobs in BFA, i’ve gotten common engi reagents from the harvester golems in kul tiras which is the most useful thing yet. In past expansion content i usually get elemental motes and ore though, not sure what is up with BFA mobs.
I’ve gotten ore from mining mobs in BfA zones. It’s not very often, but it does happen.
You likely won’t get useful mats from the mini-bar trash mobs, but you can certainly get them from large, full portrait single mobs. Get lots of Monelite from mining the azerite elementals at those Champs world quests, for example.
Usually only from mining bots with engineering. Get some wire or lite ore. But other than that no.
Engineers wanting to make the BfA engi mount have to mine a crap ton of bots in Motherlode in order to get the mats that they need. And as has been said, azerite elementals are a decent source of monelite.