Task Manager seems corrupted; Network Errors

My desktop, HP AIO PC is having "issues". This month's are : (1) Task Mngr. is missing its Column Headers (2) Services, once I've entered services, have fond what I needed I am unable to return to the main menu again of TM unless I reboot. (3) Network Error messages: 4xs per day while performing a Search (typically only with Google) I receive messages advising that website cannot be reached bc it's either down or that I am not connected to the Internet (wrong) and gives the steps to fix my DSL connection. Next, I'm advised that f it keeps happening I should talk to myself or ISP (I won't). I'm The only one using the equipment thus the System Admin. & talking to myself is a given. Running Windows 7 Home Prem. 64 bit (upgraded from Vista). How is it possible that Task Manager could become corrupted ? Yes, I use virus software which includes firewall protection. How can my system think I'm not on line when I clearly am on line ? when IE ws a constant stability issue I switched to Chrome then to Firefox, both were problematic. I'm happy with Opera & DuckDuckGo but Opera defaults to Google and that's when network errors happen. I've had 2 crashes since 2009 and have performed 2 Factory Restores. In 2014 HP forced me to do a 3rd, subsequent restore to "fix" my PC (whaaa?) & afterwards advised that my model had motherboard problems but refused to replace it bc they'd performed the restore for me and it ws also the last day of my paid, 3 -year Extended Warranty. Thieves, all of them. Could the 2 issues be motherboard related ?