To answer Q1: In fact, there is an order preserving homeomorphism between K and H. Let [a_0,b_0]=[0,1], let (r_n) enumerate the rationals in [0,1], and for each finite sequence of 0s, 1s, and 2s starting with a 0 inductively define a closed interval so that the following properties are satisfied:
a. Any interval contains uncountably many elements of K and has rational endpoints,
b. Any two intervals corresponding to sequences of the same length intersect in at most endpoints and the one with the smaller (lexicographically) sequence is to the left
c. In producing the intervals corresponding to sequences of length n+1, each interval has length at most 1/2^n and does not contain r_n as an interior point.
Then an infinite sequence of 0s,1s, and 2s starting with a 0 uniquely defines a point of K iff it’s not eventually all 0 or all 2 -- Just intersect the corresponding compact intervals of the finite subsequences, which always give a single point. The point will be a rational endpoint if eventually 0 or 2. Otherwise the intersection point isn’t ever an endpoint so can’t be rational since we avoided all rationals except for endpoints. But K is closed in the irrationals, in that if a sequence of elements of K converge to an irrational then that limiting irrational must be in K.
Finally, the same construction works for H and the desired homeomorphism just matches points with the same corresponding sequence. Continuity follows since small interval interiors go to small interval interiors.