If I were to purchase the Map Stamps Collection, would I need eighteen inventory slots?
Map Stamps do not take up any inventory space. However, the first time you buy a map stamp, you receive the World Traveler's Hat.
The World Traveler's Hat will have a falling stamp particle effect on any map you own a stamp for.
Note that only user contributed maps that are now shipped with the game have stamps, of which there are (as of 2011/11/17) 18: Turbine, Egypt, Fastlane, Steel, Watchtower, Harvest, Junction, Mountain Lab, Mann Manor, Yukon, Freight, Hoodoo, Offblast, Coldfront, Frontier, Lakeside, Nightfall, Gullywash.
Your hat also has ranks based on the number of stamps you own for a specific map. This rank changes for every 25 stamps you own for a map up until 200. Again, these are tracked separately for each map.
No. Map stamps and not stored in the backpack. They're attached to your hat but will not use up a slot.