1. All three are correct, and you've written them in the order of increasing informality.
"Begin" and "start" have the same meaning, but "begin" seems more formal in this context.
"Start" seems more neutral.
"Let's get started" is very natural and relatively informal, and it suggests that there is a job that needs to be done. You could use "start" or "begin" for a dance or song, for example, but not "Let's get started". "Let's get started" is more for tasks such as raking leaves, unpacking groceries, sorting photos and so on. It means taking the first step in a process that has to be completed.
As for the idea that 'cut the cheese " usually means " fart"/"break wind" '..... well, that's certainly news to me! I've been speaking English for a long, long time and I have NEVER heard this in my whole life. An internet search confirmed to me that this idiom does exist, but it certainly isn't universal. So no, it doesn't 'usually' mean that. But say 'cut up the cheese' if it worries you.