The fill flume is supposed to be open at the back.
The "center" pulley you refer drives the basket for spin. The transmission is driven by the large pulley seen at the left side of the pump picture above.
350367 is the 4-port pump. All belt-drive machines that use a 4-port pump should take the same one.
The lid switch shouldn't have any effect on agitation.
There are two styles of wigwag -- large-diameter (old style) plunger and small-diameter (new style) plunger. I don't know when that change occurred. Small can be retrofitted to large but the plungers also have to be changed. Unlikely large is still available, and I'd be surprised if your machine has large being that wigwags are a common failure and it likely would have been changed to small along the way.
A frozen pump will make everything run sluggish. Best not to run it that way, puts a strain on the belt.