I see you've already figured out $1$ (and hence, $3$ and $4$).
For $2$, Cantor's theorem about DLOs (dense linear orders) tells us that there is an order-preserving bijection $f:\Bbb Q\cap A\to\Bbb Q\cap B$. Note that if there is a continuous function $g:A\to B$ such that $f=g\restriction(\Bbb Q\cap A)$, then this function is unique (since $f$ is a continuous function $\Bbb Q\cap A\to B$, $\Bbb Q\cap A$ is dense in $A$, and $B$ is Hausdorff). It shouldn't be difficult to justify that there is such an extension $g$ of $f$--using methods similar to the definition of $\Bbb R$ by either Cauchy sequences or Dedekind cuts of $\Bbb Q$--nor should it be difficult to show that this extension is a bijection.
Alternately, $x\mapsto(x+1)^3$ is bijective and takes rationals to rationals, but if you want something that takes only rationals to rationals, then you'll need an approach as above.
Edit: As KReiser points out in the comments, I was far too eager to bring the big guns to the table when it wasn't necessary (that's a weakness of mine). Ignore everything I said above (though it is all accurate, apart from the claim that you need the DLO approach for a function that takes only rationals to rationals). The function $x\mapsto 1+7x$ is a homeomorphism $A\to B$ that maps $\Bbb Q\cap A$ onto $\Bbb Q\cap B$. H/T to KReiser, and apologies for the misleading answer.