What is the difference between sir, We are allowed to ride this up to three times right? and sir, We are allowed to ride this maximum of three times right? and sir, We can ride this up to three times right? ?Feel free to just provide example sentences.

The speakers internet is the same — to find out the proper instructions.

“Can I…” asks if you are able to do something.
“Can I hop on one foot? Yes, everyone can.”
“May I” asks if you have permission.
“Am I allowed” questions what the rules say.

For example, in school children can go to the bathroom during class when they ask for permission. The child says, “May I go to the restroom?” and the teacher says yes or no.

This distinction is hard because spoken language has changed. It is accepted to ask “Can I?” instead of “may I?” So you will find people use these incorrectly.

Also, you can do something if you are allowed to do it. So this is generally confusing and people do not care about nuance in day to day life.