No, you never require any doctors refer letter to avail any medical related services in any hospitals in Mumbai or India. If you have a problem and you do not understand to which doctor you must show never mind, show to the general doctor available in the hospital or casualty department and narrate your brief history to him. The doctor might treat you or possibly will refer you to the concerned specialty doctor.
Yes it always a good habit to carry old medical records along with you this will help the treating doctor to make quick decision and can avoid unnecessary investigation which will save your time and money too.
There are many reasons for doing so. If doctor feels that reports are not from qualified lab then he has every right to do so in your best interest. Your local path lab must have good standard, Should be under direct supervision of qualified M.D. pathologist. Lab must have good internal, local or external quality control program on daily basis at least one or in combination of two. For quality reporting many factor are responsible, like good instrumentation, techniques adapted, quality control, good reagents and MD doctor. No pathology lab can be operated without Qualified Doctors.