You definitly have a problem with the startup time. But i don't think it's related to your internet issue.
Are you running DHCP and DNS in the server?
Do you have Static or Dynamic IP and DNS in clients? Please post the Client and server Ip settings if you can.
when you have no internet in the server, Does your client's got internet? If client's also falls down . that means a DNS issue. Check your DNS settings in the server. If it happens every 2 -3 days. That means you have a DNs lease problem. You can extend the Dns lease.
Please let me know . Thanks