Can you fit more people in horizontal than if they stood up and you had little straps to hang onto like a subway?

In theory yes because you can use the space at the top that would otherwise be above people's heads. No guarantee they get to their destination alive though.

Once you include space to get people to the restroom though and to allow them to breath during the flight I think you get a higher density while standing vs laying down.

You can just lay them ass-to-mouth and then stack them. Come on, the slave ships and Dutch East India company taught us that one.