Electrical – run #10 and #12 cables through the same hole in framing lumber

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.

Conduit is a fine way to do it. One of the luxuries of working in conduit is you can install any wires the circuit requires. This is safer too, because it assures that hots and neutrals actually are together. There's no problem with transitioning between cable and conduit of any kind, so long as you are in locations appropriate to both types.

It's also easier to wire, since you aren't stuck with red and black (or marking white or gray neutrals). You can use any color you please; brown, orange, yellow, blue, purple or pink. Even my local lumberyard sells several of those by the foot. Of course green can never be anything but ground, even in a conduit. (thanks gregmac for reminding me to specify what I assume you know, but the next reader might not.)

In metal conduit you must bond the conduit to ground, but you also can let the conduit be the ground, so one less wire to pull. To make a transition from wired ground to conduit ground, the boxes have a hole tapped for 10-32 for attaching the ground wire. For convenience, they sell green screws 10 for a buck, and pre-made ground pigtails 10 for $3.