Multiple failures hardware/software.

Sounds like multiple issues rather than one thing being the cause.

Check the HDMI cable. There's no reason why a display cable will adversely effect your PC opening files.

Clarify "won't load sometimes". Do you mean the PC doesn't boot up? Or you've already logged into Windows but things are slow?
Related to this, what sort of drive are these files/folders/etc. on which don't show up quickly? (I'm thinking it's a mechanical drive and it's slow on spinning up. Might want to check disc health. Worst case scenario is the drive is failing.)

I suggest scanning with Malware Bytes just to eliminate the possibility of malware. Might as well be on the safe side.

Corrupted files with sfc /scannow sounds rather suspicious, especially after a reformat. Could be an issue, and personally it would be the one I would tackle first. Check for Windows updates first to see if it helps. Then perhaps look at this: http://www.thewindowsclub.com/windows-resource-protection-found-corrupt-files

To me it still fits the description of slow on spinning up. Similar to you I have an SSD boot drive, a 7200RPM HDD for games, and a 5400RPM storage drive. The mechanical drives tend not to show icons and things instantly because the drives aren't operating at the time. For my PC, I can hear the drives spin up and then see the icons load. Shortcuts are from the SSD, so I expect some are in the cache for quicker access. If it's just seconds for the drives to spin up, then it's normal behaviour from my experience. You could try defragging the mechanical drives to see if that helps speed things up.

See what fits the behaviour you see from those drives.

I would still suggest following up the other points I mentioned previously as a starting point though. An issue with the OS is never good as far as I'm concerned.

A faulty power supply will cause all of the problems you listed.
Your power supply is poor quality and has a history of known problems, including shutting computers down. I could say a great deal here, however, I do not want to confuse you with technical language.

A bad power supply can and has often destroyed other components in computers. You should have your PSU tested by a good technician.

The motherboard and attached components [RAM, etc] can be damaged already.

You can try removing all hardware except the boot drive, use only one stick of RAM, take the new video card out an put a weaker [older] video card in, and see if the reduced load lets you do more in Windows or operate longer.

I caution you here, a bad power supply can destroy components, and you have a poor quality power supply to begin with, and quality is everything. You would be much better off with a high quality 550 watt PSU than a low quality 730 Watt.

This PSU is far superior to yours: https://www.newegg.com/Product/Product.aspx?Item=N82E16817151136&ignorebbr=1

Review at jonnyguru [an outstanding review site]: http://www.jonnyguru.com/modules.php?name=NDReviews&op=Story5&reid=323

Another option: https://www.newegg.com/Product/Product.aspx?Item=N82E16817151189&ignorebbr=1

Review at jonnyguru: http://www.jonnyguru.com/modules.php?name=NDReviews&op=Story4&reid=524

Both reviews, I started with the hot testing page [hot testing is where PSUs start to show faults and why I start there], also please read the following two pages of both reviews.

Here are more power supplies if you want more wattage, with your current hardware you will not need the extra power and I do not know your budget. More wattage can help if you purchase something more power hungry in the future. https://www.newegg.com/Product/ProductList.aspx?Submit=ENE&N=100007657%20600037998%208000%20600479295%2050001697&IsNodeId=1&bop=And&Order=BESTSELLING&PageSize=60&cm_sp=Cat_Power-Supplies_1-_-VisNav-_-Certified-80-Plus-Gold-below-600w-_2

For reviews do a web search, just copy and paste the name of each PSU and add the words review www.jonnyguru.com

Jonnyguru is one of, if not the most respected PSU review sites and companies have often listened to his comments and the best companies make improvements in QC and other area.

The power supply is usually the least understood yet the most important computer component. Because the general public does not know the importance of power supplies and does little to no research they often make the wrong purchase.

If your problem started when you installed your new video card, it is also suspect and you should also have it tested by a good technician.

In your case, given I know that you have poor to questionable power supply, I highly suspect it.

It is often difficult to properly diagnose from a distance, however, the above information will serve you well.

A note about myself. I worked in the computer industry for over 30 years, mostly on the hardware side. I have to admit, I got technical and removed some of my comment here. In my effort to help people I sometimes get carried away with a lot of technical information that sometimes confuses those I try to help.

EDIT: For some reason I thought you had a fresh install of Windows. With corrupt files you need a fresh install. - Unfortunately, a bad power supply can and does cause corrupt files. It is almost a catch 22 for many people. Personally, I would purchase a new power supply and reinstall Windows.

In case you missed it, please read my EDIT at the end of my post above. Thank you.

HDMI cables are inexpensive. I keep several for my personal computer. You can get them at Wal-Mart, office supply stores and many other places.

You said you are using an extension cord. How long is it and what wire gauge?

You should be using a surge protector instead of an extension cord and replace the surge protector at least once a year. Regardless of what power cable you use make sure it is 12 gauge or you can damage your power supply, and in turn other computer components.

Extension cords should have the gauge [of the wire] printed on the extension cord or at least the packaging it comes in.

The smaller the number [gauge] the larger the wire. With computers you need a large wire in order to safely move the 20 Amps, 30 Amps, or whatever the maximum your computer will pull. It is a good idea to purchase for safety, in other words, the larger the wire the safer you will be.

Wire that is too thin will start getting warm to hot if too many Amps are pushed through it. No one wants a fire, or at the very least, no one wants a computer to fail because the power supply is pulling more amps than it should [and getting hot = shut downs or worse] because the wires going to the power supply from the wall are too small [thin].

And, the further away from a wall connection, the larger the wire you will need [for the same amount of amps being pulled].

Connections in Europe are different than in the United States and you might want to use a search engine to find guides or information about what extension cords you need.

I can simply use my eyes to determine gauge because I have done it for so many years. For modern desktop computers I use surge protectors with 12 gauge wire. Look for a surge protector with the fattest [largest round] cable.

http://www.diybyexample.info/2015/01/gauge-extension-cord/

As to extension cords in the house, for a short cord, for example 10 feet or 3 meters, 18 gauge is fine for a lamp or light or other low amp devices.

For extension cords that drive TVs use at least 16 gauge, for a long extension cord for a TV, for example 20 feet or 6 meters, use 14 gauge for safety reasons.

If you have specific questions call a reputable electrician and ask a couple of questions, do not ask too many because he will be busy and not have time for much more. Another place you can ask questions is a local fire department, as they should have an idea of what you need. Call a couple of fire departments to confirm what the person at the first fire department said, if the answers are different, use a heavier or lower gauge for safety sake. Remember, the lower the number the thicker the wire and the more amps it will safely push.