The under consideration period will be cancelled when you leave the country, no problem to leave without telling the Immigration Office that you are now able to travel.
If they used a German Passport to enter the country, the 30 day Visa Exempt Entry can be extended once with another 30 days based on tourism at a local Immigration Office.
Yes, some offices allow you during the last 45 days, others during the last 30 days of your Admitted to stay, to apply for the Extension of Stay based on over 50 years of age,
There are requirements to meet, contact your local Immigration Office for a hand out.
As there is no mention about the region where you have to apply, best option is to contact your local Immigration Office and ask for the hand out with requirements,
There are small changes, like hand drawn map or google map, depending on office.
IF you mean the big stamp in your passport with the text: Extension of Stay permitted up to xx/xx/xx and a second stamp with the text: Re-Entry Permit, you will get stamped into the country with the same Admitted up to xx/xx/xx as mentioned as permitted up to or valid until in the re-entry stamp