Minä aion yöpyä kotona huomenna.

Minä is the nominative pronoun “I.” In Finnish you almost always can drop the pronoun because the verb ending already tells you the subject (here -n in aion). So
• Aion yöpyä kotona huomenna
is perfectly natural. Including Minä adds emphasis or clarity—useful if you want to stress who is planning something.

Finnish has no separate future tense. You express future events by:
1) Using an auxiliary verb such as aikoa + main verb in the infinitive (aion lähteä = I’m going to leave).
2) Relying on context: present tense can imply future if time words are clear (Lähden huomenna = I leave tomorrow).
3) In some dialects or colloquial speech you’ll hear tulla + -maan (e.g. tulen tekemään).

Yöpyä is the basic (1st) infinitive form meaning “to spend the night.” You use yöpyä when talking about where you stay overnight (a hotel, a friend’s place, your own home).
Nukkua means “to sleep” — it focuses on the act of sleeping, not the location of your overnight stay.

• Kodissa is the inessive case (“in the home”) and is grammatically correct but sounds less natural for “at home.”
• Kotiin is the illative case (“into the home”) and implies movement toward home, not being there.

Yes: Finnish word order is fairly flexible. You can emphasize the time by placing huomenna at the start:
• Huomenna aion yöpyä kotona.
You’ll still have the same meaning, but shifting elements can subtly change the focus (time vs. subject vs. action).