Learn French: Je range mes affaires. - I am tidying my things.

It means both! While les affaires can mean "business" (as in un homme d'affaires for "a businessman"), in everyday life, mes affaires is the standard way to say "my things", "my stuff", or "my belongings".

Can I use ranger to mean "cleaning" a room?

Not exactly. Ranger specifically means "to tidy", "to organize", or "to put away" (putting things back where they belong). If you are wiping surfaces, washing, or vacuuming, you would use nettoyer (to clean).

Since affaires is a feminine noun, why do we use mes instead of ma?

Because affaires is plural here. In French, the plural "my" is always mes, regardless of whether the noun is masculine or feminine. You only need to worry about gender for singular items (mon vs ma).

The English translation says "I am tidying". Why isn't there a word for "am" in the French sentence?

French doesn't use an "am + -ing" structure for the present tense. Je range translates to both "I tidy" and "I am tidying". You can use Je suis en train de ranger to emphasize that you are in the middle of doing it right now, but the simple present tense is perfectly natural and common here.