Learn Finnish: Tiedän, että äidillä on tapana nuuhkia maitoa. - I know that Mom has a habit of sniffing milk.

This sentence uses the habitual state construction, olla tapana. Just like the standard "to have" construction in Finnish (minulla on, äidillä on), the person who "has" the habit takes the adessive case (-lla/-llä).

What is the base word for tapana, and what case is it in?

The base word is tapa (habit, custom, or way). Here, it is in the essive case (-na/-nä), which means "as a state of being" or "as a". So, äidillä on tapana literally translates to "Mother has as a habit".

Could I use haistaa instead of nuuhkia here?

You could, but the meaning shifts slightly. Haistaa means "to smell" (often a single, completed action or noticing a scent). Nuuhkia means "to sniff", which is a deliberate, repetitive, and ongoing action. Sniffing milk to see if it's still good is definitely a nuuhkia situation!

Why is the object maitoa in the partitive case?

Two reasons! First, maito is a mass noun (uncountable), and she is sniffing "some" milk. Second, the verb nuuhkia describes an ongoing, continuous action rather than a completed process, so it naturally requires a partitive object.