Also, I presume that the frame does have the braze-on mounts to attach the adjusters to? It's possible that thee frame was designed with band-on gear levers in mind, like these
In that case, you may
be successful in combining a random band with the STI adjusters, but don't bet on in.
Dito the STI adjusters should fit, unless the frame comes with on top dwontube shifter mount. In that case you have to source an extra adapter piece.
As for the headset, have you tried the simplest solution, ie tightening it? is it threaded or threadless?
There exist various size 1" headsets. Check for example Sheldon Brown for a detailed explanation. In extreme case it could be the headtube is slightly ovalized due to a crash or another reason. In that case Loctite 668 could help you solving the issue.