Wiring – Should I run conduit to protect an exposed cable and comply with code

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.

Of the options you are considering, I'd think option 2 makes more sense, and 4-5 feet of conduit - in most cases the conduit will just be continuous from the junction box right through the wall, no particular fuss "as it enters the house" - Come down the wall, put on an LB, drill a hole in the wall, insert conduit, connect to junction box, connect to LB; done. This might be aided by drilling at least a pilot hole from the inside out to make lining it all up easier. Also pick up some duct seal (gray putty, in the electrical aisle) to pack the wires as they leave the LB.

If the wires are entering via an existing conduit, you just need to connect a new conduit to the box that conduit terminates in, and your junction box.

EMT or PVC is six on the one hand, half a dozen the other. I use metal aboveground and PVC below, mostly.