Learn Finnish: Koira menee huoneeseen. - The dog goes into the room.

This is the "into" form (called the illative case). For most nouns, you make this form by doubling the last vowel and adding -n (like talo -> taloon). But words that end in -e, like huone, take a special ending: -eseen. So huone becomes huoneeseen.

Why is the verb menee spelled with a double "e" at the end?

The dictionary form of "to go" is mennä. When talking about "he", "she", or "it" (like the dog), you take the verb's stem (mene-) and double the last vowel. That makes it menee.

Could this also mean "The dog is going into the room"?

Yes! Finnish doesn't have a separate verb form for things happening right now (like the English "is doing"). Koira menee translates to both "The dog goes" and "The dog is going" depending on the context.

If we added the adjective pieni (small) to describe the room, how does it match the noun?

In Finnish, an adjective must match its noun in both number (singular/plural) and form. Because huoneeseen is singular and in the "into" form, pieni must do exactly the same, becoming pieneen. If the dog went into multiple small rooms, both would change to match the plural "into" form: Koira menee pieniin huoneisiin.