I understand how being randomly disconnected can be a frustrating thing to deal with. When it comes to your streaming being disconnected you want to look into what's going on with the router. It could be that the bandwidth isn't strong enough to support a constant stream or it could also be the network speeds that you are connecting to.
With the 350 model you want to be at the minimum of 2mbps upload & 2mbps download as well. That speed will give you a sustainable picture & video. Below is a couple of links that will further aid you in testing your speeds, an article that will give you some steps for streaming troubleshooting & an article that shows you the tech specs for the 350.
Keep in mind that the article for streaming issues is labeled as troubleshooting for the Solo but because the 350 plugs in to the network the same exact way as the Solo you could reference these steps.
I hope this information helps resolve your issue. Feel free to respond with any further questions!
The Sling Moderation Team