På badet er det en liten lekkasje under badekaret, og jeg blir urolig.

På badet means “in the bathroom” in a location sense (where something is happening/located). Norwegian often uses på with rooms/places as a general “at/in” location.
You can sometimes hear i badet, but på badet is very common and natural for “in the bathroom (area/room).”

This is the existential construction: Det er … = “There is/are …”.
Here det is a dummy subject (it doesn’t refer to anything specific). The real new information comes after the verb: en liten lekkasje.

Badet is the definite form, meaning “the bathroom.” In context, it’s typically a specific bathroom (e.g., in your home).
Norwegian often uses the definite form where English might just say “in the bathroom” without emphasizing “the.”

Badekaret is the definite form: “under the bathtub.”
Norwegian often prefers the definite form when referring to a specific, known object in the situation. Under badekar would sound incomplete or like a generic category (“under bathtubs”).

The ending -et is the usual definite singular ending for neuter nouns.
Here the base noun is et badekar (a bathtub), so the definite is badekaret (the bathtub).

Yes: badekar = bade (bathe) + kar (tub/vessel). Norwegian forms compounds very freely, and the last part decides the grammatical gender.
Since kar is neuter (et kar), badekar is neuter (et badekar).

Bli means “to become / to get.” So jeg blir urolig = “I get worried / I become uneasy” (a change of state).
Jeg er urolig would mean “I am worried/uneasy” (describing your current state, not the change into it).

There’s a comma because two independent main clauses are joined: 1) På badet er det en liten lekkasje under badekaret
2) jeg blir urolig
After og, the second clause is still a normal main clause. Since it starts with jeg (the subject), the verb follows normally: jeg blir (no inversion needed).