That "connection reset by peer" is an indication of the connection being terminated in an unexpected to Galileo way. That may indicate a problem with the networking port. Try plugging it into the same port where you had Yun (which worked fine). Galileo is very stable platform where it pertains to networking, so these issues you're seeing are unlikely related to the board itself, I would say they're rather caused by the network. First of all the DHCP is apparently setup incorrectly (or is intentionally giving out wrong data, who knows, but you overcome this with the Google's DNS) and that "connection reset by peer" may be an indication of a physical level problems.
Please post the "ifconfig -a" command results for both Yun and Galileo, let's see if there any packet drops there.