Which elements of the fundamental group of a surface can be represented by embedded curves?

Here's an answer to the second and third questions.

The third question has a general answer: In any group, the kernel of the abelianization (i.e. the set of homologically trivial elements of the group) is generated by the conjugates of the commutators of any generating set.

For the second question, the claim you reported reading is fuzzy and hence inaccurate. The correct statement is that if an element $\gamma \in \pi_1(F)$ can be represented by a homologically nontrivial simple closed curve then there exists an automorphism $\Phi : \pi_1(F) \to \pi_1(F)$ such that $\Phi(\gamma)$ is one of the standard generators.

To prove this, first, each of the standard generators is represented by a nonseparating simple closed curve.

Second, the simple closed curve representing $\gamma$ is nonseparating (otherwise the homology class is trivial).

Third, for any two nonseparating simple closed curves $c,d$ passing through the base point there exists a homemorphism $\phi : F \to F$ taking base point to base point such that $\phi(c)=d$. This is true because $F-c$ and $F-d$ are homeomorphic (an easy appplication of the classification of surfaces), and a homeomorphism between them can be found which extends over $c$ and $d$ to be a homeomorphism of $F$ that takes base point to base point.

Finally, the desired automorphism $\Phi$ is represented by the homeomorphism $\phi$.

You are looking for the work of Birman and Series I believe. Here is a paper they wrote which looks at this for simple closed curves on orientable surfaces with boundary. I know that there is a paper for surfaces without boundary, also by Birman and Series, which came out after this one, but I cannot find it at the moment. I will look around and let you know if I find the other paper, but maybe someone else knows where it is.

They give an algorithm which lets you determine if a word in $\pi_1(F)$ represents a simple closed curve on the surface.