"I swim every month."
Tsuki, gatsu, and getsu are different pronunciations of the same kanji and generally mean month or moon. I think the main pronunciation is based off of context and the other kanji in the phrase. It appears that gatsu is used mostly as a suffix but the other two are used based on the specific word or phrase and mostly may need to be memorized?
I don't think you can because technically the subject of this sentence is "I" and the meaning would change significantly by making "every month" the subject. Japanese often drops everything but the most important parts of a sentence so it is not unusual to see sentences like this.