Yes, do note that if its a pre-planned operation, you may also get the agent help to inform the insurance company to issue the GL in advance.

This is particularly useful if the admission was during the holidays.

Also, do note that GL issued can be cancelled at your request. GL issued by the insurer may also be cancelled by the insurer if there is an evidence of miss declaration of the medical condition.

For example, during policy inception, the medical condition was not declared OR the medical condition is found to be of congenital issue, eg sinusitis due to narrow nasal passage.

I believe the admission fees are depends on Insurance company & policy. Some insurance company will no need admission fees/co-insurance.

With GL letter, hospital will directly bill to the insurance company. Any exclusion or clause will state in the GL letter.
Of cause GL letter only issued when the hospital is their panel. Else the person will need to pay & claim which are quite troublesome.