Learn Finnish: Poika liukastui ja kaatui. - The boy slipped and fell down.

Both are Verb Type 1. First, find the stem by dropping the final -a from the infinitive, giving us liukastu- and kaatu-. Because these stems end in -u, we simply attach the past tense marker -i to create the -ui ending. Finally, because the subject is poika (he), we don't add any personal ending.

Why isn't the final vowel doubled in the third-person past tense, like it is in the present tense?

In the present tense, the third person ("he/she/it") doubles the last vowel of the stem (e.g., poika kaatuu). However, in the past tense, the rule changes: you only add the past marker -i and stop. You never double the -i for the third person in the past tense, which is why it is simply kaatui.

I remember the verb kaatua changing to a "d" (like in kaadun). Why does it keep the "t" in kaatui?

This is due to consonant gradation. The weak grade (d) appears when a personal ending "closes" the syllable with a consonant, like the -n in minä kaadun. The third-person past tense form kaatui ends in a vowel, leaving the syllable "open". Open syllables take the strong grade, so the strong t remains!

Does kaatua work for any kind of falling?