Learn Danish: Min nye kat løber ind i skabet. - My new cat runs into the cabinet.

In Danish, i often just means "in" (describing a fixed location). By combining the directional word ind (inwards) with i, we get ind i, which means "into". It emphasizes the motion of the cat moving from the outside to the inside.

Could I use inde i instead of ind i?

Not for this exact meaning. ind i shows a change of location (motion into the cabinet). If the cat was already inside the cabinet and just running around in circles in there, you would use the stationary form inde i (inside) instead.

Why does the adjective ny have an -e on the end (nye) if it describes just one cat?

Whenever you use a possessive word like min (my) right before an adjective, that adjective must take its weak (definite) form. For almost all Danish adjectives, this means adding an -e. So even though it is a single cat, ny becomes nye.

Why does the word skab have -et added to the end?

The word for cabinet or closet is et skab, which is a neuter noun (a t-word). To say "the cabinet", you attach the neuter definite ending -et to the end of the word, making it skabet.