Learn Finnish: Hän avasi oven koputtamatta. - He opened the door without knocking.

It comes from the verb koputtaa (to knock). To make the 'without doing' form, take the strong verb stem (koputta-) and add -matta (or -mättä, depending on vowel harmony). This special ending directly translates to 'without knocking'.

Can I just use the word ilman (without) here, like ilman koputtaa?

No, ilman is mainly used before nouns, like ilman takkia (without a coat). When you want to express 'without doing an action', Finnish usually avoids ilman and instead attaches this -matta or -mättä ending directly to the verb.

Why is the door oven (with an -n) and not just ovi?

oven is the object of the sentence. Because he completely opened one specific door, it takes the total object form, which gets an -n ending (looking just like the genitive case). If the action were somehow incomplete or ongoing, it would take the partitive case (ovea).

What is the dictionary form of avasi, and why does it look so different?

The dictionary form is avata (to open), which is a type 4 verb. A tricky rule for type 4 verbs is that in the past tense, their -t- changes to an -s-. That is why avata becomes avasi for the hän (he/she) form.