Procedure to automatically solve field theorems in a SMT solver
All of the theorems you want to prove (except uniqueAddNeutral) are of the form $\forall x . \phi(x)$ where $\phi(x)$ is quantifier-free.  Herbrandization is typically the first step towards proving those theorems: replace $x$ with an arbitrary constant $c$, with no assumptions made about $x$, and prove $\phi(c)$ holds without making any assumptions about $c$.

The theorems you want to prove are simple enough that exhaustive search over candidate proof trees might suffice to find a proof.  At each step, you apply one of the axioms, instantiating the universally-quantified variables in the axiom with appropriately chosen values.  You have to choose which axiom to apply and how to instantiate the variables; typically you instantiate each variable in the axiom with one of the subexpressions in the current expression/goal, so there are only finitely many choices at each step, and you can simply exhaustively search over all possibilities (up to a given size/depth).