NetTechGuy

This might be of interest if you want a Dremel - like tool, it seems that they are now making a compact screwdriver: https://www.dremel.com/en_US/products/-/show-product/tools/go-01 I believe this is available at Lowe's and Home Depot stores in the US, though I do not recall the sell price and have not tried one myself. Otherwise, DeWalt and Makita both make compact cordless screwdrivers of a similar design, though they will be more expensive than this one. Happy shopping!

Have you tried a shorter cable? HDMI is fussy, and has a distance limitation of around 10 meters using a passive cable. I would suggest first trying a different cable, if the issue persists, then reduce the length and the issue will likely resolve itself. If you still need to go 10 meters, you might do well to look for what is called an active cable, which amplifies the signal and helps mitigate issues such as this one.

The Ubiquiti Unifi line is pretty good if you want to get away from an all-in-one device, something like an 8 port or better PoE switch to feed an AP or 2 depending on the size of your home is a setup our clients often go with that delivers good performance. If you drop in a Unifi USG as well, that will give you your routing as well as a nice hardware firewall while staying in the same product family. Configuration is pretty easy, and the products are widely available in the US market. Good Luck!

I would suggest a wall mount bracket designed for mounting 19" network equipment in the proper orientation, they are not terribly expensive and available in different depths and mounting capacities depending on the size of the equipment. If the unit weighs more than a couple of pounds, get a small piece of plywood to put behind it and the wall if the surface is drywall or plaster to prevent the unit falling off. Hit up amazon or somewhere similar and search for patch panel wall bracket, you will find all kinds of good options at various price ranges.

Wiring looks fine on this end, do you have a switch to plug into? In order to make the Ethernet active, you need a network switch and some patch cables to run from the ports at the top of the termination hub (this panel) to the ports on the switch/ router/ modem. The panel shown above is simply a termination point for cabling, it does not have any electronics to communicate with your devices. As a quick test, plug an Ethernet cable from the LAN port on the modem into the RJ45 port above your new cable, then plug a laptop into the wall plate on the other end and see if you can get to the internet.

This sounds like an IP address conflict, most likely your devices are receiving or using the same address that they are getting from the university switch, which is why they work individually but not when connected to the unmanaged switch you purchased. You could also be using a static IP, which can be determined by looking at the IP settings for your network adapter on the PC or PS4. If you want both devices to be connected to the internet at the same time, you would most likely need to purchase a router instead of the switch. Connect the internet (WAN) port of the router to the wall jack, then connect your PC and PS4 to the LAN ports, and configure the router to assign IP addresses via DHCP. If the connection to the PC is using a static IP, that would be set as the default gateway for the router.

Based on the above, I would have to agree with the others that the internal battery is likely dead or damaged. I would find it unlikely that the internal battery was removed, but the X series laptops are designed so that they discharge the removable battery first, then the internal once the external is depleted. This allows you to hot swap the external battery without the system shutting down. If the prior owner left the laptop plugged in most of the time or rarely depleted the external battery, the internal is probably degraded due to non-use. You could always check the internal battery connection, but I would suspect it is time for a new battery.

As long as you use a quality coaxial splitter you shouldn't see any performance degradation. Ideally, you would use a splitter designed for digital cable or satellite signals, stay away from the cheap ones and make sure to install terminators on any unused ports to prevent signal bleed.