I don't know why it might be disallowed by local codes. As mentioned in the comments to the question, this stuff is often put into poured walls and floors. It is used quite heavily for 240V wiring and phone/data wiring in Israel where almost all construction is concrete. We also used it for both 120V/240V and data in WaterShed here in the USA.
As another answer mentioned though, it's not good for outside use.
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.
Best Answer
Yes, this is a good idea, but a few details.
No more than four cables per conduit, unless you're willing to upsize ALL the cables to the next larger size. That is 310.15(B)(3)(a).
The conduit needs to be fairly large, because the cables are oval. Each oval cable is treated like a single wire of the wide dimension (because they twist). They can't fill the conduit more than 30% (for two) or 40% (for three or more). 1-1/4" conduit is probably a safe choice, though 1-1/2" will be easier to pull. (pulling stiff Romex is a bugbear). *This is not a Code requirement per se, but it'll surely be a practical limit given NM-B tends to have kinks, and kinks snag like crazy in conduit.
Flex is a bad idea. I know it seems easy to install, but the flex will fight you every inch of the way. The energy you are using to push or pull the cable will instead bend the conduit sideways. Believe me, this is very frustrating, especially when pushing, and pushing is (otherwise) easier since you only have to watch one end. With a non-flexible conduit secured at both ends, that's an non-issue.
Assemble the conduit first, then fish the wires through it.