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 Jiří L. Fajstrup and J. Rosický.
The second question was already answered in the comments: if $G\colon \mathbf{Top} mathbf{Top}^{\mathrm{op}} \rightarrow \mathbf{Set}$ is continuous, then it has a left adjoint $F$ by the special adjoint functor theorem. Therefore we have natural isomorphisms
$G(X) \cong \mathbf{Set}(\ast,GX) \cong \mathbf{Top}(F(\ast),X)$,mathbf{Top}^{\mathrm{op}}(F(\ast),X)=\mathbf{Top}(X,F(\ast))$, which shows that$G$is represented by$F(\ast)$. Edit: added the missing op's mentioned in the comment. 1 The category of topological spaces is not locally$\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$-generated spaces is locally presentable can be found this paper of Jiří Rosický. The second question was already answered in the comments: if$G\colon \mathbf{Top} \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}(F(\ast),X)$, which shows that$G$is represented by$F(\ast)\$.