Competitive equilibrium with production
First, note that both firms have constant returns to scale. This implies that firms must make zero profit in equilibrium. If they could make a positive profit, they could make even more profit by scaling up production and no profit-maximizing production plan would exist. So who owns which firm does not really matter. We know from the first welfare theorem that every equilibrium must be Pareto efficient. With the given utility functions, this means that both firms have to produce a positive amount. But this pins down the relative prices of $x$ and $z$, and of $y$ and $w$, respectively. So this reduces the problem to one in which you have to find two prices. You can set one of the remaining prices to be $1$, and then calculate the excess demand for one good. FInd the price that makes this excess demand zero. As a consequence of Walras' law, this price will clear all markets.