Is this rejection rude? Should I apply to another position in the same University?
This seems to me like a normal rejection letter.  They normally don't really contain any information except "you didn't get the job".  Some letters do go on for longer about "we had so many great candidates and we wish we could hire them all, blah blah blah" but I wouldn't really say that makes them more polite.

In particular, I certainly would not draw any conclusions about the "healthiness" of the environment from this letter.  It might be a good department or a bad one, but this letter doesn't help you to determine which; you'll need to investigate that in other ways.  See:


How can I detect a toxic environment while on the interview circuit?
&quot;Warning signs&quot; that candidates should be aware of during an academic job interview
What things can I do as a job market candidate to gauge the internal political culture of a department that is hiring?