Modern coax for cableTV/Satellite/outside antennas/Cable modems is RG6; it has better shielding and a larger center conductor thus a bit less signal loss.

As said, you may have RG59 (from back in the days before cable modems) which may work for a cable modem, but is less than optimal. Try it, if it works for you then it's your lucky day. If you have low speeds or lots of errors, the RG59 coax cable in your walls could be the culprit. There's also RG11, but I doubt you will find it in a home. RG59/RG6/RG1 explained.

RG59 is about 6.3mm (1/4 inch) OD (outside diameter), RG6 is about 6.9mm (9/32 inch) OD. Comparison between those two.

Another thing... if you have no idea of what is along the cable run feeding that little junction you pictured, then you don't know if there is a signal splitter somewhere along the way. In the old days of outside TV antennas and primitive cable TV, they would often use splitters to distribute the signal to various rooms.

Today, the standard is to pull a "home run" of coax cable from each room/drop all of the way back to a central point in the home - likely the basement, if you have one. Splitters could be hiding in the attic, or behind walls, but that was generally bad practice - they should have been place somewhere visible.

A splitter is a no-no as you basically lose at least 50% of your signal for every splitter along the way. Splitters are/were for TV signals, but they aren't friends with cable modems.

Finally... the funny little connectors you see usually use a 7/16ths inch wrench, and are called "F" connectors.

Finally... a cheapie tone generator + probe may keep you from going crazy if you have to chase out cables, but don't use on lines with AC power. More expensive ones may come with other doodads and be a higher quality... people that use them everyday spend more for good ones.

Huh? Home computer fans are 99.97% of the time DC.

For years, I've used a "wall wart" type supply to run a computer fan for various needs; just make sure the amps that it can supply are equal to, or greater than the amps that the fan will draw.

FYI... I have a bin full of "wall warts" to choose from - when I get rid of old gear, I always keep the power warts so I will always have a handy supply. Solder on a different end, if needed, heat shrink tubing to finish the job right.