Is this a sound logical syllogism about evolution being intelligent?

It's not valid.  It violates the fallacy of (the) undistributed middle   

P2 says Intelligence emerges from networks, but doesn't say it always emerges from networks.  Therefore you can't conclude that because evolution is a network that evolution is intelligent  ("network"" is the term that's undistributed).

P4 doesn't relate to the other premises. It seems to be saying that evolution is intelligent because it solves external problems, but then concluding that evolution is an intelligent process is begging the question 

If you are asking us if it is sound , then this is outside of the scope of this site and I would assume the expertise of the members here, myself included. So just note that as far as the soundness goes (truth of the premises) you will be getting opinions.

As for just the validity, I would say "no" due to the wording. Simply claiming "this is how intelligence works" in the premise cannot bring us to a "therefore... x is an intelligent process" in the conclusion. A good syllogism uses the same terminology in the premises and conclusion.

I suspect there is more wrong here, but I will leave it at that.