20fifi wrote: IB to ESA transfer: Is it best to mention DLA award on ESA50 form or not? 
It is entirely up to you, but it can do no harm.
A DLA HRM award will be supportive of of your mobiliing (question 1), however the ESA test is far stricter, so it will only be indicative of problems and will not result in an award.
A DLA Care award may indicate a a level of disability, but as it is no-specific, it will only provide general support to your ESA claim.
Gordon