Is it possible to calculate the random factor $r$ from a encrypted message and the private key in a Paillier cryptosystem?
It is strange that Wikipedia propose to choose $r\mod N^2$ while $r^N\mod N^2$ depends on $r\mod N$ only:
$$(r+tN)^N=r^N+r^{N-1}tN^2+\ldots\equiv r^N\pmod{ N^2}.$$ It means that you can recover only $r\mod N.$ In order to do it you can use the formula from the cited answer
$$r\equiv (r^N)^M\pmod{ N}, $$
where $M = N^{-1}\bmod \phi(N)$.