Is it online or on-line?
AP says: "online" (one word, no hyphen)
When do we use online program? When do we use online degree? When do we use online degree program?
"Online program" and "online degree program" could be synonymous, except that we might offer an "online certificate program" (no degree). Use "online program" when talking about a program delivered via online delivery system. Call a program an "online program" (same with "online course") only if it is 100% online in terms of delivery. Anything other than that, call it "hybrid."
Do we use online college or online campus?
We would say "online campus." We have an Undergraduate College and Graduate College now. The other terms we use are "undergraduate evening program" and "undergraduate traditional program" (no capital letters). Since we’re delivering programming that is either undergraduate or graduate already through online delivery, say “online campus”—given that campus refers to a place and online is a virtual place.
Do we call it William Woods online or WWU online?
You would say "William Woods University Online." You could use "WWU Online" for a nickname, but we don’t use WWU in any of our official names, since you don’t find it online if you search William Woods," (you get Western Washington University).