If I don’t find a job by July 15th, could I transition from "job seeker visa" into a "tourist visa" (valid for 90 days)? I know that I can not extend the job seeker visa, but I was wondering if I could just change the type of visa. Or do I have to leave the country for that?
This would not exactly be a visa, as a US citizen you are allowed to stay for 90 days anywhere in the Schengen area without a visa. Since you don't need (and cannot get) a short-stay visa, you don't have to apply for it, pay a fee, or complete any formalities abroad. If you weren't already in the country, all you would have to do is presenting yourself to the border.
Now, the rules are not super-clear about the need to leave the country, it is not exactly required but some countries do recommend doing it (and I think Germany does but I am not sure). Note that to get an entry stamp you do not need to go to the US, only out of the Schengen area (so the UK, Ireland, or Croatia would do for example). On the other hand, going to a neighbouring Schengen country like France or Poland is not enough.
If you are planning a short trip to the UK to get a stamp, it's important to remember that entry in the country is never guaranteed. Border guards are empowered to check your eligibility and to deny entry for a number of reasons.
If I am on a "tourist visa" and I receive an offer letter from my future employer, could I request the work visa/permit? I already have a residence permit and tax ID (because of my job seeker visa) and my job might allow me to apply for the Blue Card.
Yes, you can do that. It's usually forbidden but US citizens (and citizens from a handful of other countries) are exempt from this restriction and can in fact apply for a residence permit from within Germany without a long-stay visa. You are not allowed to start working while waiting for the decision, though.
