Guten tag

Sorry, but your post is unclear in various respects, thus my reply below based on assumptions. Maybe you can explain what your real plan is?

1. You need a marriage visa only if you want to stay on in Germany after your wedding, since the marriage visa (unlike all other visa types) can be converted to a residence permit.
How long it takes depends on the embassy, and how long you need to furnish the required documents. It can be several months.

2. I have no idea what PSA is. But if the embassy needs birth certificates you better provide them - or explain the situation to them and ask if there is any other way.

3.  Which investigation officer? What would he/she investigate? Knowing German buerocrats, I doubt that they'd do anything on their own - rather, they would ask you to submit proof of everything you say.

Altogether, as you can see in various related threads, marrying in Germany as a foreigner is very difficult and many advise to do it elsewhere (recognition of a foreign marriage is easier).