I'm thinking of making like a "dog run" on the side of my house. Like a place where it can go to the bathroom,

Don't put in anything that could be easily destroyed (like your umbrella idea). Get a hard plastic kiddie pool for cooling off or playing. You might want to try both the tarp and the dog house. Some dogs like houses and some don't, and some just like them because they're cozy and don't think of them as a place to get out of the weather. Keep a very close eye on the dog at first to find out if he's trying to escape by digging, jumping or squeezing through a space you thought was too small to fit through.