ct.category theory - Local presentability and representable presheaves over the category of topological spaces

The category of topological spaces is not locally $lambda$-presentable for any $lambda$. The reason for this is the existence of spaces which aren't $lambda$-presentable (a.k.a. $lambda$-small) for any $lambda$ (in a locally presentable category every object is $lambda$-presentable for some $lambda$). An example of such a space is the Sierpinski space; a proof of this can be found in Mark Hovey's book on model categories, on page 49.

There is a convenient category of topological spaces which is locally presentable, the category of $Delta$-generated spaces. This category contains most of the spaces usually studied by algebraic topologists (for example, the geometric realization of any simplicial set is a $Delta$-generated space). Daniel Dugger has some expository notes on this here. A proof that the category of $Delta$-generated spaces is locally presentable can be found this paper of L. Fajstrup and J. Rosický.

The second question was already answered in the comments: if $Gcolon mathbf{Top}^{mathrm{op}} rightarrow mathbf{Set}$ is continuous, then it has a left adjoint $F$ by the special adjoint functor theorem. Therefore we have natural isomorphisms

$G(X) cong mathbf{Set}(ast,GX) cong mathbf{Top}^{mathrm{op}}(F(ast),X)=mathbf{Top}(X,F(ast))$,

which shows that $G$ is represented by $F(ast)$.

Edit: added the missing op's mentioned in the comment.