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.
Your options are to engineer a new path or to open up the drywall around the places where the spray foam was used. This shouldn't be a major problem as drywall is easily patched and refinished.
Even with finding a new path you may very well have to open up access holes to allow for drilling through fire blocks or the top/bottom plates at floor transitions.
Best Answer
Just to update with an answer: at the end only one cable was necessary, which helps. Solution still applicable if two cables were necessary (as originally stated): a plastic raceway of the right size is best in this case. I checked dimensions of all raceways I could find, and chose the one right the size I needed. That avoid extra wasted space and a tripping hazard. The photo might not tell that, but even with two cables it would not be a problem, it wouldn't be projected too far from the door to be unsafe.
The result is nice: