What happens, when an unlimited bandpass-signal gets mixed?
what happens, if the RF-signal from the antenna isn't band- or lowpass-filtered?


If the system is analog, lots of uncontrolled harmonics. If the system is digital, then yes, you will get aliasing.

Aliasing itself is not a form of mixing. But sampling is.

When an analog signal $x(t)$ is sampled at distinct times $\frac{1}{Fs}$ seconds apart, it is effectively mixed with a train of impulses at frequency $Fs$. Sample-and-hold turns these spikes into "steps". This is what is causing the "mirroring" of the spectrum. 

In fact, by tuning your $Fs$ carefully, you can "shift" a signal into a baseband by undersampling it.


  I know for example from decimation (lowpass-filtering and downsampling) that there would be aliasing, which isn't a form of mixing but nonetheless.


Probably it is clear by now that it is. Take an analog signal $x(t)$, multiply it with a train of impulses $u_s(t)$ at some frequency $Fs$, now multiply the result by another train of impulses $u_d(t)$ that is equivalent to "taking out" every other pulse. If you don't first low pass $x(t)$ for the spectrum you are trying to "squeeze it in" post mixing by $u_s(t) \cdot u_d(t)$, you will get aliasing. It's the same concept applied twice.

Hope this helps.