Are Submergence tasks available to SRM players to help reduce the karma cost of Submergence? Likewise, if we are a member of a Network, do we get the discount there too?

I belive so yes,

As long as you don't need a GM, and can fit it into the downtime rules that are in place, I don't see why not. Unfortunately, I'm not seeing a lot of submergence tasks that fit that description. Source Code, maybe. This is just off the top of my head, mind you; nothing official.

Maybe you could pose the question for the upcoming Shadowrun chat, which features Shadowrun Missions. It would give Stephen and I a chance to look the question over and come up with an official ruling.