Yes, both are fine. The 'down' just adds additional information. You could say 'up' as well - perhaps for someone like a surf lifeguard sitting in a very tall chair. We also use 'sit up' to mean 'sit up straight' instead of slouching, and 'sit up at the table' to refer to sitting somewhat formally at a table for a meal. I'm not sure of the relative ages for a baby being able to understand instructions, but usually they always need help to get into a high chair! Yes, you are quite right about the directions in your last question. (edit - sorry, I missed the fall bit) - again, the 'down' adds more information. But here, usually the 'down' suggests 'completely'. You can 'fall' slightly, such as a trip, or stumble. But 'fall down' means you end up in a pile on the ground.

Come/go you are right. Sit down is the action of lowering yourself into a seat. Once you have sat down. You are then sitting. You can sit someone down, like a baby. In both cases, we use down to express the lowering action. With fall down, you fall to the surface you were standing on. In this case, fall and fall down are interchangeable.

As for the kitchen question, yes, you are exactly correct.

As far as "sit" does we say that all the time instead of "sit down." I would argue that sit down is often used as a command--if you want to harsher with someone you might tell them to "sit down!" Whereas you might invite someone to "have a seat." "Sit" is somewhere in the middle. With most of these tone of voice is what matter more than the actual words. "Fall" and "fall down" are similar, but you can't always use fall down. "Fall" is used when things go downward accidentally or due to gravity. "Prices fell today" or "I saw the pie fall off the table." "Fall down" is used when things go downward accidentally or due to gravity and hit the ground. "I fell down off my bike and hurt my leg." You cannot say "Prices fell down today."" For "fall down" it needs to be an object that can hit the ground. Come to the kitchen / Let's go to the kitchen. You are correct!

Hello Viktoriya, I love this question. Short answer: the commands to 'sit on' and 'sit down on' are basically the same. It's strange, but the word 'down' really doesn't change the meaning. Even stranger, suppose the person obeys the command, to 'sit down on the chair,' but sits down with poor posture, slouching forward, or leaning back against the chair. The same person who told them to 'sit down' might say 'Sit up!' which is a command to sit down, but with a straight back. Fall and Fall down are not as close. For example, you can 'fall in love,' but you cannot 'fall down in love.' So, if a guy really likes a girl, we might say 'he fell,' or 'he fell for her.' You can also use it to mean that someone was tricked, like 'he fell for that salesman's lies.'