Learn Finnish: Onko tuo äidin ompelema takki? - Is that the coat sewn by mother?

When you use the agent participle (like ompelema) to say who did the action, the doer of that action must be in the genitive case. So, mother (äiti) becomes äidin.

How is the word ompelema formed?

It is the agent participle of the verb ommella (to sew). You form it by taking the strong verb stem, which is ompele-, and adding the participle ending -ma.

Could I say this using a joka (which/that) relative clause instead?

Yes! You could say Onko tuo takki, jonka äiti ompeli? (Is that the coat that mother sewed?). The agent participle construction äidin ompelema takki is just a more compact way to say the exact same thing, and it is very common in written Finnish.

If we were talking about multiple coats, would ompelema change?

Yes, the agent participle acts like an adjective, meaning it must agree with the noun it describes in both number and case. If you were asking about multiple coats (takit), the participle would take the plural ending too, becoming Onko nuo äidin ompelemat takit?