At the moment I only have an HF receiver (or I will when I build it, hopefully soon). Are there any contests I can participate in, or ways for me to participate in a contest without borrowing a transmitter or working with an amateur who has one?
Summits on the Air (SOTA) is not a contest, but an ongoing activity with awards. You can participate as an activator (transmitting from summits), a chaser (working those on summits), or an SWL (listening to contacts and logging them). All the chaser awards are available to SWLs (shortwave listeners).
More info is here: http://www.sota.org.uk/Awards
Other activities also have SWL awards. There appears to be one for IOTA (Islands on the Air).
This site is dedicated to "shortwave amateur radio listening" and has a news about awards and a registry: http://swarl.org/
There is a complete list of all Amateur contests at: http://www.hornucopia.com/. Some ham contests include a category for SWLs, others don't. Check the contest's rules at this site to see if includes an SWL category.
Absolutely, you can be a SWL (Short Wave Listener). Here are some links to be a better SWL: