Surge protectors are little more than fancy and expensive extension cords. All they do is chop off "clamp" the tops of sinewaves of most surges and spikes. Extreme surges and spike result in the protector (
IF functioning properly) cutting power - instantly crashing the computer, potentially corrupting the data (and operating system) on your drives.
Surge and spike protectors do absolutely NOTHING for "low voltage" events like sags (opposite of surges), dips/dropouts (opposite of spikes), or long duration sags (brownouts) - all of which can result in instability and system crashes.
For these reasons, ALL computers should be on a "good" UPS with AVR - automatic voltage regulation. A "good" UPS with AVR will protect all your computer and network equipment from all sorts of powerline anomalies originating from "the grid" and also, from other high wattage devices in your home (like refrigerators, toasters, and $15 1500W hair dryers made in some obscure factory by indentured, often under-aged workers using parts from a sister factory up-river).
And note until now, I have not mentioned anything about battery backup during total power outages.
Another problem with surge and spike protectors is they wear out. They are like motorcycle helmets. If a helmet saved your noggin once after hitting a curb, it did its job and you need to get a new helmet - even if the old helmet still "looks" okay.
A nice 1000VA or larger, "good" UPS with AVR should easily be able to support your computer, TV/Monitor, and all your network gear. I prefer APC. I have a 1500VA APC UPS supporting my i7, 16Gb computer, modem, router, switch, plus two 22" monitors.
I keep saying "good" UPS because like power supplies, there are cheap UPS and good UPS. Don't get a cheap one. They don't have as sophisticated line monitoring or regulation as the better UPS and the resulting waveform may be too "dirty" for the devices connected to them. Also, a cheap UPS may not have a fast enough "cut-over" in the event the batteries need to take over.
As for your 600W PSU, note a computer only draws from the PSU what the computer needs. So if your computer (motherboard, RAM, drives, etc.) only needs 250W, the computer will draw from the PSU 250W, regardless if the PSU is a 600W PSU or 1000W. And the PSU will draw from the wall just 250W, plus another 35-75W due to PSU inefficiencies (energy wasted in the form of heat).
If you think your old surge protector may be a problem, by-pass it for now.
The fact your memory test showed no errors is good. But sadly, RAM can test good by these software based testers and still not work when put into use. If the tester reports errors, the RAM is bad. But if no errors reported, it might still be bad. So to conclusively test, you need to run with just one stick of RAM at a time, and see of problems develop. If no problems, swap sticks again until you [hopefully] find a bad stick.