A Birkhoff variety is a class of algebras closed under division and arbitrary products, a pseudovariety is a class of algebras closed under division and finite products.
Now for each type of variety, if it is generated by a finite number of algebras, then their universal algebra for a fixed set of generatos $A$ (i.e. the algebra over which each map $\varphi : A \to M$ with $M$ belonging to the variety could be uniquely extended to a homomorphism from the free algebra to $M$) is finite too, but not in general.
For a Birkhoff variety the universal algebra always belongs to the variety, for a pseudovariety this is not the case (the universal algebra is a profinite object). Now I am looking for a simple proof of this fact, more specifically how does for example the universal algbra (w.r.t. to a generator set $A$) for the pseudovariety of all finite groups look like?
I am not sure I understand your question correctly. If a variety of monoids is generated by a finite number of finite monoids, then every finitely generated relatively free monoid is finite, but the converse is not true. Take for instance the variety of idempotent monoids which is not finitely generated...
As for free profinite groups, there is a huge literature on the subject.