Learn Finnish: Kirjoittamalla parannan mielialaani.

This sentence demonstrates pronoun drop (pro-drop), which is the focus of your current lesson. Because the noun mielialaani already ends with the 1st person possessive suffix -ni (meaning 'my'), the separate pronoun minun can be safely dropped.

Is it also normal to drop the subject minä (I) from this sentence?

Yes! Just like with possessive suffixes, you can usually drop 1st and 2nd person subject pronouns. The verb parannan ends in -n, which already clearly means 'I improve'. You would only include minä if you wanted to emphasize that you are doing it, rather than someone else.

How does the word kirjoittamalla translate to 'by writing'?

This is formed using the third infinitive (the -ma / -mä form) combined with the adessive case ending (-lla / -llä). When you put these together, it describes the means or method of doing something. So, kirjoittaa (to write) becomes kirjoittamalla (by writing).

Why is the word for mood (mieliala) in the partitive case before the possessive suffix is added?