Is every countable dense subset of $\mathbb R$ ambiently homeomorphic to $\mathbb Q$
Two countable totally ordered, densely ordered sets without endpoints are isomorphic---this is a theorem of Cantor (Gesammelte Ahbandlungen, chp. 9, page 303 ff. Springer, 1932) Thus two countable dense subsets of $\mathbb R$ are homeomorphic, since their topology is induced by their orders.
Now, if $A$, $B\subset\mathbb R$ are countable dense subsets, fix an order isomrphism $f:A\to B$ and extend it by continuity. What you get is an homeomorphism $\mathbb R\to\mathbb R$.