I want to install an outlet right under a light switch and not have it controlled by the switch. Can I wire it from the switch?
Also, instead can I use a combination switch/outlet in the box where the present switch is so a second box isn't needed?

If a Neutral is in the box, you can. If you only have a black and a white, you can't.
If in conduit, you could push or pull in another wire(a neutral).
You can put a combination device here.