Avarage classes for PP (probabilistic polynomial time) and PPT machines running in expected polytime
(This is a more detailed version of my comment on the question.)

As for your question 1, your class is R, the class of decidable problems.  Expected polynomial time is not a useful resource restriction when you allow unbounded error.

To prove that your class contains R, consider a language L∈R.  Let M be a Turing machine for L which halts on every input.  Then we construct a probabilistic Turing machine M′ which runs in expected polynomial time as follows:


Simulate one step of M.  If M accepts, then accept.  If M rejects, then reject.  If M has not halted, continue.
Halt and accept with probability 1/4, halt and reject with probability 1/4, and go to step 1 with probability 1/2.


It should be easy to see that if M halts after t steps, then M′ surely halts before executing step 1 more than t times, and that the probability of error is at most 1/2−1/2t &lt; 1/2.  The expected number of times step 1 is executed is 2(1−1/2t) &lt; 2, and therefore M′ runs in expected constant time.

To prove that your class is contained in R, first note that finite expected running time implies that the probabilistic Turing machine halts with probability 1, and therefore your requirement implies that the machine halts with the correct answer with probability greater than 1/2.  Simulate the probabilistic Turing machine, accumulating the probability of acceptance p and the probability of rejection q.  Because the machine gives the correct answer with probability greater than 1/2, either p or q exceeds 1/2 eventually, and at this point we can stop the simulation and give the correct answer.  (For more details, see this answer by Karolina Sołtys.)