Your picture isn't working... but yes, water will come out the exhaust.
On that note........
The constant alarm is over temp. AND, generally, a garden hose doesn't have enough flow to keep your engine cool. So, I hope you didn't run it long with the alarm going off.
As far as the high idle... Get into the side of the engine and make sure the linkage isn't binding. You should be at 1000~1100 RPM.