fall apart means break into pieces (e.g. My new dress fell apart in the washing machine.)
fall down means fall to the ground (e.g. The picture that you hung up last night fell down this morning.)
fall out (with some body) means stop being friends (e.g. They used to be very good friends. I’m surprised to hear that they have fallen out)
figure something out (e.g. I need to figure out how to fit the piano and the bookshelf in this room.)
fill in a form, a questionnaire means writes the necessary information on a form. (e.g. Please fill in the application form and send it to us by 25 April) (= fill out)
fill something up means fill to the top (e.g. I always fill the water jug up when it is empty.)