1) and 2) H1B is cap-exempt only if she had been on an approved H1 before. If this is teh first time she is apllying for H1 and if rejected, she can contest it or apply in next years quota. One option is to continue working in EAD (if its valid) till next April and apply again.
However she can look for opportunities in certain insitutions like Universities, NGO's who are cap exempt and can apply for H1B all through the year, which will be the other option if she need H1 immediately.
3) Is MTR for denied visa necessary in any of the above cases?
MTR is not necessary as far as I know if you have your L2 still valid to be in status. You can continue in your previous status i.e. L2. If your previous status is expiring you need to do apply for extension or MTR to maintain your status.
It is always advisable to consult your immigration lawyer to be on the safe side.