In general, for randomized classes complete problems tend to be either promise problems or approximation problems (which means they don't technically satisfy the conditions for being complete problems).
If you allow approximation problems, you can get complete problems in BPP. For example, for BPP you can ask: given a Turing machine $M$, approximate the probability that it accepts within $t$ steps on input $x$, where $t$ is polynomial in the size of the input. It's clear that you can approximate this probability with a BPP machine (using simulation), and it's also obvious that any BPP language can be reduced to this problem. Technically, this problem isn't in BPP since it's not a language (i.e., its output is not $\{0,1\}$), so it's not BPP-complete.
You can turn this into a promise problem by imposing the "promise" that the acceptance probability either be greater than $\frac{2}{3}$ or less than $\frac{1}{3}$.
These complete promise problems or approximation problems play the same role that complete problems play for non-randomized languages, and they really deserve more respect from computer scientists. For quantum computing, the natural complete problems also tend to be approximation (or promise) problems, and they have been quite useful in the theory of quantum computing.
There is a natural promise problem from quantum computing (stoquastic Hamiltonian) which is MA-complete (and not trivially so). However, I don't believe there are any languages (non-promise problems with $\{0,1\}$ answers) known to be MA-complete.
It is not known that $PP \subsetneq PSPACE$, but it is natural to conjecture it. What people really believe on this subject is murky territory. The fact that $P^{PP}$ contains the polynomial hierarchy is, to me, evidence that $P^{PP} = PSPACE$.
Let $a(n)$ be an unbounded time-constructible function. One example of a class which is "between" the polynomial hierarchy and $PSPACE$ is $\Sigma_{a(n)} P$, the class of languages recognized by an alternating machine that makes $O(a(n))$ alternations on inputs of length $n$. It is not known for example if $\Sigma_{a(n)} P \subseteq P^{PP}$ for any unbounded $a(n)$. If you were going to prove that $P^{PP} = PSPACE$, you would want to start by putting some "unbounded levels of the polynomial hierarchy" in $P^{PP}$ first. However, it may be helpful to know that by doing so, you would separate $PP$ from $LOGSPACE$. For all unbounded $a(n)$, we have $LOGSPACE \subsetneq \Sigma_{a(n)} P$; this was proved in
Lance Fortnow: Time-Space Tradeoffs for Satisfiability. J. Comput. Syst. Sci. 60(2): 337-353 (2000)
Best Answer
See the Complexity Zoo, which seems to aim at being current. It currently lists 495 classes, including PPAD, as you mention, and a large diagram of some of the classes.