Can I run multiple wires through conduit for a hot tub and an outlet?

Generally if all wires are line voltage the answer is yes. Low voltage and line voltage wires are not allowed in the same conduit. Fill requirements apply, so the size of the conduit is dependent upon the number of wires and gauge of each. Tables exist to determine the correct conduit size. But IMHO, it's always best to go with larger conduit than required for a couple of reasons: 1) It can be easier to pull the wires and 2) you might need to add more for some other purpose later.

EDIT: One more thought: It doesn't matter that you have 120 and 240 in the same conduit, since in actuality 240 is really just 2 120 volt lines 180 degrees out of phase.

You can, yes (provided you respect conduit fill limits), but usually there's an easier way. Since hot tubs need a local disconnect anyway, you can use a small subpanel (often called a spa panel) to supply both the hot tub and the outlet, and that will also serve as the disconnect. Just make sure to get one with more than two breaker spaces.

Here's an example of one:

Simply add a second breaker for your outlet. If you anticipate using some high-current load through the outlet at the same time the subpanel is on, you'll want to size the subpanel's feeder wire somewhat larger (e.g. use a >=70A feed for a 50A spa + 20A outlet). If you'll only be plugging light loads into the outlet, or you can turn the spa heater off when the outlet is in use on high current loads, then you don't need to bother up-sizing the feeder.