Why is this a requirement?
Well, if $r$ has (say) $p$ as a factor, then it makes the unblinding step rather difficult.
The original $m$ has $N = pq$ possible, as does $m^d$; if $r$ is a multiple of $p$, then $mr^e$ has only $q$ possible values; if we pass it to the signer, it'll come back to us with a value $m'^d$ that also has only $q$ possible values. We cannot map $m'^d$ back to $m^d$ (as a single $m'^d$ value would map back to $p$ possible $m^d$ values, and we don't know which one it is.
A follow up a related question: after $s$ (unblinded signature) and $m$ (original message) are revealed, can anyone (and not only the signing authority) check the validity of the signature by checking $s^e=m \bmod N$?
Yes, anyone with the public key can validate it. This is generally true for public key signature algorithms.