Hello,
I work downtown Calgary. I would not generally recommend driving a camper downtown. I rarely drive my truck downtown and spaces are tight. There are a few spots which are above ground which don't have overhead limitations. I'm 6'7" with my canopy on and I don't fit in 90% of parkades, obviously you will be taller but I'm just giving you a general idea. Parking is insanely pricey. You will generally not find anything cheaper than $20 per day. Expect $30.
The West side of town is much nicer than the East. As another poster mentioned, look into travelling via C-Train. The West arm is pretty decent to get into downtown but parking at the station may be a little limited. The NW arm is pretty good to commute into and there is some great camping spots N-NW-W of town. There is also the south leg of the train. It is a much longer commute into downtown, expect at least 30 minutes from the very southern station. There is also some great spots south of town to stay overnight. It is also very easy to drive to the southern most station and park. Heads up, it looks like there is tons of open space to camp and explore just west of town, it is Native reserve, pretty much a no-go-zone.
May is sure a variable month for weather. I've had a week of +25C and a week of +5C. You may get a dump of snow but it will likely be gone in a day. You should expect rain. May and June are the rainy months in Southern Alberta. On average, I would expect 15C during the day as mentioned.
PM me if you would like more information. Spots to stay, decent restaurants, a good pub or two downtown, local attractions etc.
Roamers of the back woods
2006 Ram Megacab SRW. 19.5" Visions
2010 Eagle Cap 850 - current
2005 2500 9.4 Bigfoot - sold
2000 Hawk Four Wheel Camper - sold