Learn Italian: Finiscila con questi lamenti! - Cut it out with these complaints!

It doesn't refer to any specific feminine noun. It is a fixed pronoun that belongs to the lexicalized verb finirla (to stop it / to cut it out). Verbs of cessation like finirla, smetterla, and piantarla all use this empty la to represent 'it' or the current annoying situation you want stopped.

Why is it followed by the preposition con instead of di?

Because the thing being stopped is a noun (questi lamenti). When verbs like finirla or smetterla are followed by a noun, they require the preposition con. If you wanted to express stopping an action (a verb), you would use di + infinitive instead, for example: Finiscila di lamentarti! (Stop complaining!).

How is the imperative formed here, and why isn't the pronoun before the verb?

This is the informal tu command. In affirmative informal commands, object pronouns attach directly to the end of the verb (finisci + la = finiscila). If this were a formal Lei command, the pronoun would have to come before the verb instead: La finisca con questi lamenti!

What is the difference between finirla, smetterla, and piantarla?