So I didn't end up testing anything this past weekend. I was on the computer playing games most of the weekend and didn't use the internet a great deal, but I never saw the status meter loose connection with the modem all weekend. When I used the browser the pages were sometimes pretty slow to load at first, but I didn't have any DNS/server unresponsive timeout messages. Right now it's working pretty much just like it should.
I have my router setup for Google's DNS servers right now. I used them in the past without any improvement, so I hesitate to proclaim this is what fixed my timeout issues. If it is, then that really doesn't speak well for the HughesNet IT staff. I didn't Traceroute it but I'd guess I added 3-5 hops into each lookup. Latency grows with each hop, unless Hughes has serious internal network infrastructure problems. I'm more inclined to think I had a good weekend and my service quality will degrade again soon. I can live with it if it works more often than it doesn't, I don't have very high expectations, I have been using HughesNet for a long time.
As for the modem issues, I am at a loss. Nothing I have done should have effected that. The only possible explanation I can come up with is that being new out of the box the modem needed some software updates. Maybe over the first few days the modem was trying to update it's software and that caused it to be unresponsive and reboot. I can't really say if this theory is possible, I know the modem's OS is a Linux variant and by and large they update without rebooting/lockup issues (usually just restart a service and your done), but I have no idea if any updates to the software were made.
I'm just happy it appears to be stable for now. Unless it starts acting up again, I am content to live with it as it is.
I appreciate the assistance, and if my problems return I will post here again. I would like to say that I'm very happy with the support from this forum. If the rest of customer support was as good as this, I would be a much happier customer.