Whats the big problem with getting your log book stamped? Just about everyone has managed to get it done.
They might accept a certified print out of hours if you dont want your log book stamped, however best check first as it goes to the DGCA.
Im fairly certain it needs to be from your current/last employer.
Time from application to contact seems to vary, it depends on how well qualified you are re the current vacancies. If you are type rated you should hear quickly if not then it may take a while!.