1.What could have caused a desktop comp to completely shut down. It just all of the sudden, one day, wouldn't turn on.
***Well, the 1st things that come to mind is maybe a power issue, virus/malware, hard drive died or something got fried. The best way to find out is to take it to a friend or a shop to see what they say is the problem.
Whatever is wrong may be replaceable/fixable.
What were you/your friend doing when it last turned on? What happened?
2. Also, what could I do to get the memory from it or have I pretty much lost it all?
***If you're talking about memory, you could open it & take it out. If you mean like rescuing your files, the best thing may be to take it into a shop to see if they can recover them.
There are programs you could use, but the hard drive would need to have power in order for them to be used. (Which means you'd probably have to take it out & put it in a working desktop or external hard drive enclosure & then use the program.)
The best thing would be to take it into a shop or send it to a place that specializes in data recovery.
3. Another thing is, I have a High Speed Internet from Verizon(DSL Modem) and it goes out every time it thunders. Confused This is really, really weird. Do you know why?
***I'm not sure. I think maybe what you could do is call Verizon, file a support ticket & ask them to come check it, just to make sure nothing is damaged. Is there an antenna outside where you are? Maybe the signal is getting disrupted, but it should be able to withstand some thunder.... (& even lightning)
How is it plugged up? Is it and your computer on a surge protector?
4. Finally, if you look up all of your websites through search engines does that protect you from viruses and other stuff to a degree?
***Using a search engine doesn't usually give you a virus, unless it's been attacked or something or it's just not reputable. It's when you click on the link to take you from the search engine results to the site. You can run into problems when you go to a site directly.
There is a search engine called Startpage.
http://www.startpage.com/ or
https://www.startpage.com/
When you do a search on Startpage, there is a word usually under each result called "Proxy". If you use something like that, then that's a way of reducing stuff getting on your computer.
The other ways are to have antivirus software, free or paid, especially if it's Windows & some anti-malware/anti-spyware programs that you use to scan your computer.
I hope this helped! If you have more questions, feel free to ask.

I'm not an expert, but I'll do my best to assist!