Lower bounds on monotone space complexity
This not quite an answer, but rather an "extended comment".

To question (i): property (*) holds because of read-once, not because of the monotonicity, and this is why the argument fails in the read-many (even monotone) case: then the combined paths need not be PMs, they can be arbitrary graphs containing a PM.  

In the read-once case, even the Exact PM (accept a graph iff it is a perfect matching) requires exponential size (by the same argument as yours). But if we allow negations, then EPM can be computed by a circuit of size $O(n^3)$: check if every node in A has degree at least one, and if every node in B has degree at most one. The resulting switching network is "almost read-once": every consistent path is read -once. Search term "null-path" here for more information.  

To question (ii): I haven't understood what "it" here refers to? But as far as I know, the $n^{\Omega(\log n)}$ lower bound of Razborov (for monotone circuit size of PM) remains the strongest one. Even though monotone switching networks constitute a special case of monotone circuits (where one input of each AND gate must be a variable), no stronger lower bounds for PM are known here.