Do some/most golf courses offer a discount of sorts to other people who work for/on golf courses or are in the golf field?
For the most part, yes, at least as of 1 year ago. But professional courtesy goes both directions. Don't call and ask for a comp round when a particular course is in its peak season. So, don't call a course in Florida in December, don't call a course in Connecticut in June, don't call a course in Pinehurst in April. But just about every course will let another professional play the course for free if times are open; and especially if you are playing with 2 or 3 paying customers.
When I was in Florida during January-April 2008, only once did I, or anyone else in my PGA program, call and ask for any kind of deal. And that was when 15 of us wanted to play together on the same afternoon; we worked it out with the local muni to get on for cart fee. Every other time (well, we just naturally got a deal at PGA Golf Club, but that was not so much 'asking a favor' as much as they knew we were there and knew to give us the normal deal) I just payed the rack rate.