Single player is not defined entirely by the amount of human players, since it is possible in some source ports to play in cooperative or deathmatch mode while being alone, possibly with the addition of computer bots.
Competition is possible in single player through demo recording and speedrunning. The old "Doom Honorific Titles" required a player to demonstrate the ability to complete levels in single player mode while fulfilling specific conditions.
In vanilla Doom and Heretic, things appear in single player mode unless the "multiplayer only" flag is set. Boom added the "not deathmatch" and "not cooperative" thing flags, allowing to have things that only appear in single player mode. Hexen introduced a different system where flags are explicitly set to mark a thing as appearing in single player, deathmatch, and cooperative game modes.
When the player dies in this game mode, the level is reset and the player has to start it again from a pistol start. In Hexen, however, the last save made is reloaded instead, because the hub-based level design means that a reset of the level and the character's inventory might make the game unwinnable. (For example, losing a key or puzzle item picked up in another level might prevent unlocking access to the next hub.)