If you are on the ground floor, 'downstairs' means the basement or cellar. The words are not synonyms in all cases.

In the U.S., no. For us, "downstairs" is normally the ground floor, also called the "first" floor, the one that has the main entrance. "Upstairs" is usually the second floor. "Cellar" is synonymous with "basement". However, as Claire said, if someone on the ground floor mentions "downstairs", logic impels the conclusion that they are talking about the basement, because there is nothing else that is "down" from there.

Not necessarily. While "downstairs" can sometimes be used to refer to a cellar, it typically refers to the floor of a building that is below the main level, and may include basements, sub-basements, and other below-ground levels in addition to cellars. A cellar, on the other hand, specifically refers to a room or set of rooms below the ground level of a building that is primarily used for storage of food, wine, or other goods. So while "downstairs" can sometimes be used to refer to a cellar, it is not necessarily synonymous with it, as it can refer to other below-ground levels as well.

Everybody is over thinking this one. Generally the surrounding context would tell us. And in daily conversations the second person can could would ask for clarification as to what the other was referring to. Also we get used to friends neighbours/neighbors way of speaking.

Yes no problem.