Electrical – run electrical cable and sound cable in the same conduit

HDMI and speaker wire are basically signal wires, not power wires. Both the voltage and amperage is very low. Indoors it is safe to run these hidden or exposed without channel or conduit if they are properly rated. See this Q&A for a discussion of proper rating.

When channel or conduit is used, it is either for convenience in handling, and to keep the wires away from other materials, or it is used to improve the aesthetics - channel can be painted and looks neater than cables.

There are limits as to how much signal wire can be run in cable or conduit, but this is generally a function of pure physical space and ability to pull wire, rather than heat or safety. Speaker wire can be tacked to baseboards or other molding if that works. be careful to avoid breaking the insulation or the inner wires when stapling, and if you put wire under moldings, be careful not to nail through. Its not a danger issue, is a broken wire/no signal issue.

Power, including extensions to run a projector, is very different. No code allows extension cords to be buried in wall. I don't think any codes allow extensions to be tacked to baseboards. Also power cables should not run parallel to signal wires - they can cause interference even though many signal wires (like HDMI) are shielded.

You need to bring a regular, properly installed power line to the projector. This can be properly installed NM cable inside a wall to an outlet box. It can be surface wiring, but this requires standard metalic surface channels and boxes such as this and this:

One other note. Lampcord is often used as speaker wire. It is the same as is used for AC power cords and small extension cords. If you are using such wire, be careful, if you are burying in walls or stapling, that it is speaker wire rather than power carrying wire.

If you simply used the ohm meter to check between the wire pair at each of the three terminus points then it is very likely that you would read infinite. The reason being is that the other ends of the pair are laying open someplace else.

To check for the wire routing you would need to check for continuity from one wire at say point C0 to one of the wires at point F.

The two pairs of wires at point F may have been connected to speakers located at or near that point. If that is the case then it is likely that all six of your speaker wires terminate at some location that you have not yet located. Some systems that people install put all the equipment out of site in a closet or adjacent room. This is particularly true for cases where the equipment is almost totally controlled through a user interface on the video screen.

You may have wire pairs looking something like this coming from a closet someplace:

You could investigate trying to trace the wire routing. There are tools that you can purchase that inject a signal into a wire from a battery operated transmitter. Then there is a separate receiver unit with a probe that you slide along the wall. The receiver module will produce a sound that chirps and varies in intensity depending upon how close to the driven wire it is located the probe. I own several of these units and have good to mixed results with them. Here is a link to the unit that I have had reasonable success using.

You can find this unit at better hardware stores, big box locations or online. The price showing at the link above is decidedly better than what I paid for the same unit about 10 years ago.