Is venting PC exhaust into a wall cavity a bad idea?

I wouldn't dump it directly into the cavity. Warm indoor air can carry moisture, and if it reaches a cold enough surface inside the wall, condensation is possible. That doesn't mean mold is guaranteed, but a wall is a lousy place to discover a moisture problem late.

A continuous duct through the wall is different from using the cavity itself as the air path. If you want to explore that, check the wall construction and local requirements first. Insulation and sealing may be needed, depending on where the duct goes.

You'll move some heat out of the room, not out of the house. The hallway and cooling system still have to deal with it. Also check where your case actually exhausts. A duct on one fan won't capture much heat coming out through other vents.

Before cutting into a wall, try reducing the heat the PC makes. A frame rate cap, lower power limits, or an undervolt can cut power use without necessarily making games feel worse. How much you save depends on the hardware and settings.

If the room is still too warm, better airflow between rooms or additional cooling may be more useful than a duct that only catches part of the exhaust.

An opening between rooms can act as a transfer path, but the size of a PC fan doesn't tell you whether the wall opening will move enough air. A proper transfer grille is worth considering separately from a ducted PC exhaust. Check for anything inside the wall and any fire or building requirements before making openings.