Is every group a Galois group?

No. Profinite groups are residually finite (in fact a group is residually finite if and only if it embeds into its profinite completion) and many groups are not residually finite. If you don't have any particular restriction on the number of generators, $\mathbb{Q}$ is a simple example.

There are other restrictions. A compact Hausdorff group has a Haar measure of total measure $1$ and no countable group can be equipped with such a measure since the measure of any singleton cannot be $0$ and cannot be positive. This rules out $\mathbb{Q}$ but it also rules out, for example, $\mathbb{Z}$.

This cannot be true since a profinite group cannot be countable (see this) so therefore we cannot make $\mathbb{Z}$ into a profinite group.