You are to focused on the text of law, instead of the actual implications for the person needing a abortion.
If you look at the actual law: https://www.gesetze-im-internet.de/stgb/BJNR001270871.html#B...
It boils down to (per paragraph):
(1) if you follow the right procedure (consulting, 3 days waiting then decision, done by doctor, no more then 12 weeks) it basically doesn't count as a crime (practically it's in a semi state between crime and not crime due to the constitutional court ruling you mentioned, it's anyway clear cut something you can not be convicted/pursed for, which is all that matters).
(2) if there are serve medical issues abortion is not against the law. This paragraph can theoretically apply to some cases of rape victims as it does explicitly mention mental medical issues/damage additional to physical issues.
(3) explicitly/automatically makes 2 apply to rape victims through only up to 12 weeks (since getting pregnant, not since the last period). Doesn't mean it can't apply post 12 weeks.
(4) the person having the aborting can not be punished if the abortion was done by a doctor and no more then 22 weeks passed (the doctor might be punished thou), if the person was pressured into it the judge can decide to not punish even if they normally would need to.
So what does that mean
- no the government is not involved, you never have to ask the government, only doctors
- yes post 12week there is no automatic exception, but you still do not need to ask the government a doctor is enough
- through theoretically a court case could be pushed to decide if the decision of the doctor was correct and paragraph 2 does apply to a rape victim in a post 12 week abortion so the best way to avoid that is to make sure its very clear that it's the case before doing the abortion, the lead to the common practice of getting two doctors to agree with the abortion (one of which isn't involved in the abortion in any other way). Even better (but not required!) would be to get the agreement of a state doctor (which is I guess where the idea of needing government approval comes from). Through saying a "state doctor" is part of the government is very very misleading, technically true trough. But their role is more like "doctors the government believes are competent and trust". I.e. their medical decisions are still independent from the government.
And sure this can be improved by a lot.
But then at least harassment of people going for a abortion is neither common nor tolerated in Germany (but seems to be common in parts of the US). Similar there is no in general no "attack" against doctors which do abortions, nodded of abortions as reasonable in Germany. Which from what I have heard happens in some US states often enough to be called systematic.
In the end for any conclusion I guess you would need to speak with people having gone through abortions (in various contexts/circumstances/US states/German states etc.).
PS: Depending on when you start counting the 12 weeks here might also be referred to as 14 weeks.