Contribution came from Emmanuel Vantieghem.
Q1 : there are such numbers. The smallest example is W = 836 and N
= 421 (it is prime, but not a prime greater than s(W) = 844 ; I guessed
N is not required to be composite).
Q2 : take W = 836 and N = 839.
I found many weird numbers w for which the smallest prime p that
makes w p is weird is much smaller than w. For instance w = 7192,
where p = 31. In general, it is possible to prove a stronger result
than that of puzzle 599: if w is a weird number and p a prime bigger
than the sum of all the divisors of w, then w p^n is weird for all n
> 0. (could it be a challenge for some puzzlers to recover my proof ?)