Learn Hardcore Norwegian: Hun finner tannbørsten og tannkremen på badet. - She finds the toothbrush and the toothpaste in the bathroom.

To form the present tense you drop the –e from the infinitive and add –er:
å finne → finn + er = finner.
Unlike English, Norwegian verbs never change their ending for person or number, so you use finner for jeg, du, han, hun, vi, dere and de.

Why are tannbørsten and tannkremen in the definite form, and how would the sentence change if you used indefinite forms?

Because you’re talking about specific items (the toothbrush and the toothpaste), each noun takes the definite suffix –en (common gender).

Indefinite: tannbørste (a toothbrush), tannkrem (toothpaste)
Definite: tannbørsten (the toothbrush), tannkremen (the toothpaste)
If you said
Hun finner tannbørste og tannkrem på badet.
it would mean “She finds a toothbrush and toothpaste in the bathroom,” i.e. unspecified items.

Why is the location expressed as på badet instead of i badet, and what does på badet literally translate to?

In Norwegian certain rooms and locations (kitchen, bathroom, loft, etc.) use på rather than i. Literally på badet is “on the bathroom,” but idiomatically it means “in the bathroom.” It’s simply the standard preposition choice for that room.

What is the gender of bad in Norwegian, and how do you form its indefinite and definite forms?

Bad is a neuter noun.

Indefinite singular: et bad (a bathroom)
Definite singular: badet (the bathroom)
Neuter nouns take the suffix –et in the definite form.

How would you ask “Where does she find the toothbrush and the toothpaste?” in Norwegian, and how does the word order change?

Begin with the question word Hvor, then the verb, then the subject. The rest follows:
Hvor finner hun tannbørsten og tannkremen på badet?
(Literally: “Where finds she the toothbrush and the toothpaste in the bathroom?”)

Can you move på badet to the beginning for emphasis, and what effect does that have on word order?

Yes. Norwegian is a V2-language, so if you front an adverbial like på badet, the finite verb must stay in second position, causing subject–verb inversion:
På badet finner hun tannbørsten og tannkremen.
This emphasizes “in the bathroom” at the start.

Are there other words you can use for “the bathroom” in Norwegian, and how would the sentence look?

Yes. A common alternative is baderom (neuter):

Indefinite: et baderom, definite: baderommet.
You could say:
Hun finner tannbørsten og tannkremen på baderommet.
Another word is toalett → toalettet, but that often refers more narrowly to the toilet itself.