The issues are:
1. Bottom bracket. The one recommended for your BB shell width should be OK, but be aware that there are (a) chainline issues and (b) Q issues.
2. Derailleur. Yes, adjust down, and you might find, depending on the BB spindle width, that you will be at the absolute limit of the derailleur cages outward movement.
3. Q or tread. MTB cranks are shaped so they are further away from the seat tube than road cranks. This is to accommodate the wider chainstays of MTBs. You will (a) need to decide if you like the wider foot placement when you ride (b) and need to be aware that this can alter your biomechanics such that you can strain knee ligaments until you are used to it.
I have a Fuji Touring that came with road triple cranks, and I swapped them out for square taper 22-32-44 MTB cranks. I had to fiddle with the BB to get the Q or tread equal on both sides for the BB I was using. The original Tiagra triple derailleur was kept and remains on there, but as I mentioned in (2) above, it was near its outer limit when shifting to the large ring.
I have now replaced that 22-32-44 with the 26-36-48 Deore MTB crankset. The derailleur is the same, but the BB was the Octalink version, and the standard MTB spindle width for my BB shell has produced a good chainline and not extended the derailleur so much.