Connection Failure and Spotty ISP

Hello. I am by no means a computer wiz, but manage to do OK with manual configurations. Most of the time. Two days ago, I had an emergency computer replacement (shattered the screen on my MacBook and was quoted a Tier 4 repair - $$$). I managed to migrate my files using a combination of the migration assistant and file sharing via Ethernet between my new MacBook Pro and the old machine. Everything seems a-ok with one major annoyance - I am getting connection failure prompts every time I boot up and every 30-60 minutes, my ISP fails.

More details:

1) When I boot my MacBook Pro, I receive the following prompt: "Connection failure: The server "(name)'s MacBook" may not exist or is unavailable at this time. Check the server name or IP address, check your network connection, and then try again."

Why is my current computer trying to connect with my former? I am assuming this has something to do with how my home network is configured. I looked at my router settings and cannot seem to ID where I would need to make an adjustment.

2) When I experience an ISP failure (going through Network Diagnostics), I don't have to do anything other than prompt the "Assist me" and the issue seems to repair on its own.

Will a router reboot correct this issue (I hate reconfiguring security, etc.)? Or, is this a larger AirPort problem?

3) I called my ISP and they deferred me to Linksys (router manufacturer), but I cannot help but wonder why this would be a router issue...

Perhaps the connection failure prompt and ISP issues are unrelated? I appreciate any assistance and I apologize in advance for knowing/understanding all tech-speak.

Thanks!