Learn Finnish: Koira tulee sisään.

tulla is a Type 3 verb, which ends in two consonants and a vowel (like -lla). To find the stem, you remove the final -la to get tul-, and then add an -e- to make the stem tule-. For the third-person singular (he/she/it) form, you double the last vowel of the stem, resulting in tulee.

How would I say that the dog does not come in?

You need the negative word ei for the third-person singular, followed by the verb stem without any endings. Since the stem of tulla is tule-, the negative form is ei tule. The whole sentence becomes Koira ei tule sisään.

Why is the word for 'in' sisään here, rather than sisällä?

Finnish has different words for location depending on whether there is movement. sisään means 'into' and is used here because the dog is moving into a space. If the dog were already resting indoors, you would use sisällä (inside) because there is no movement to a new place.