I sent an inquiry for a PhD position at a German university and she responded on the 30th of July to submit a research proposal (without a deadline). I sent her the paper on the 19th of August. Do you think that I can continue with the search and consider it as a rejection if she doesn't respond to the research proposal after a few days?
Yes, you should continue with a search as always until you secure something.
No, it doesn't imply rejection and it would be a mistake to assume so, especially in the summer when some professors take vacations (Germany, perhaps, doubly so) and others are involved in professional activities (travel, obsessive writing, ...)
And, in general a "few days" of academic time is just a microsecond.
You will hear something specific eventually if the system you are in (Germany) involves individual professors in the application process. In the US, you might not, however, since applications are to departments, not individuals, and handled by committee. Patience is advised, but keep your options open.