Has anyone on the support side been able to explain just what a "stuck process" or "stuck table" is? I presume something has gotten into an infinite loop. For a database, it would seem to be an error in the MySQL code (nothing you can normally do about it, unless you're doing some very advanced procedure stuff), but for a process, it might be something in your code. I suppose that it could also be some sort of deadlock over resource contention, but again, that's almost always an OS issue and not end-user code.
Has support offered any guidance on what might need updating in .htaccess or php.ini when your server is upgraded or account moved? Some obsolete settings may cause a 500 error, but I'm worried that from the sound of it, there are old settings that are simply ignored, or worse, do the wrong thing now. That should be the responsibility of the hosting company (LP) to at least document such things, if not scan customer accounts.