(1) It is a theorem of Stallings [1978, "Constructions of fibred knots and links"] that closures of positive braids are always fibered links. Thus "most" knots are not realised as the closure of a positive braid.
(2) If you allow positive (say) crossings only, and TL generators, then you can use the latter to "rotate" the former by 90 degrees. Then the given diagram can be converted without too high a "cost".
(3) Interesting question. I'll guess that the blow up is at worst polynomial.