ASTs and SSA are orthogonal concepts. ASTs are a representation of syntax (including SSA) where immediate subterms are easily accessible (unlike with strings as representation). Easy access to subterms is important for the efficiency of type-checkers and code-generators. In contrast, SSA is a simple programming language that is often used as an intermediate form in code generators.

HOAS is a form of embedding a language L with binders into a language L-Base also with binders and functions, but such that L-binders are handled by functions in L-Base and the problem of defining capture-avoiding substitution for L-terms is reduced to reusing L-Base's capture-avoiding substitution. (NB L = L-Base is possible but not required.)

HOAS is neat, but (simplifying a bit) prevents you from making inductive definitions on L terms. There are various ways of getting around this shortcoming, the most principled one is probably the nominal approach of A. Pitts et al. Another problem with HOAS is that the representation may contain 'junk'. That means there are L-Base terms that do not represent L-programs.

HOAS representations can only contain junk if you have not taken appropriate precautions. See the paper "Boxes Go Bananas" for one technique.

Yes, that's right. But you need parametric polymorphism for "Boxes Go Bananas" which may not be available in some languages.