Probably you have two or more phone jacks daisy-chained together. One cable goes to wherever the phone line enters the house and the other runs to another phone jack somewhere else in the house.
This is a common practice. I did some pricing online, and it seems that Cat 5e is comparable in price to 4-conductor phone cable, so the builders may simply have used it because they had it to hand.
Well, your pictures have twigged an occasional issue I run into where I can't see them, (likely not your fault) but flying blind....
From a functional point of view you really don't have to worry about separation. Twisted pair is actually quite good at ignoring noise, and 60 Hz noise is of little note to 100MHz ethernet anyway. You can do it all wrong and it will work, 99% of the time. I don't suggest that you do it all wrong, I do suggest that you don't freak out about it.
From a code (and safety) point of view you should not have low and line voltage going through the same hole, and from a hyper-cautious network standpoint they should be separated by a good 12" when parallel, or cross at 90 degrees if they need to cross. Separation matters a lot more in an industrial environment with noisy devices on the powerlines than in a typical single-family residence.
From a "best practices" point of view, network and power in separate stud bays (when running vertical) or between different sets of joists when running parallel to the joists is certainly a best practice, though not required by code - it maintains separation quite aggressively.
Notches are far worse than "holes in the center third" (top to bottom) of a beam, joist, or header. Holes should not be too close to the ends, even in that center third. I'm not sure there's any need for the cables to go through the header if they can "float by" (remember, I'm flying blind here)
Portable electric heaters are nasty, nasty loads. If you are plugging one into a 15 amp (14 Ga) circuit that, all by itself, with nothing else on, is loading the circuit very near to maximum (if it's a 1500w heater, actually more than is permitted for fixed/hardwired loads that might (as with a heater) be on for more than 3 hours. A warm wire is fully expected under those conditions. A 20 amp (12Ga) circuit would be more appropriate if you are in the habit of using one.
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This was your original BELL system wiring throughout your house. It would have been modified throughout the years as bell added new cable for the old DSL service. You can see the old service drop coming in from the top and the distribution phone lines branching out from there. Who are you with now? If you're not using any of that wiring for your existing service, you can use it for whatever you need. I know this is vague but so is the picture.