do, while loop examples.. I'm struggling with..

You're right, the variable is overwritten.

So looking at your first example:

let number = 10;

do {
  number = prompt("What time do you go to bed?");
} while ( number !== "10" );

alert("You can go to bed now!");

1. the variable number is declared and assigned the number 10
2. the do statement is executed. Note that we are not checking the while condition yet. The do statement will always run at least one time since it ignores the while condition the first time. So it doesn't matter that the number 10 (which is the current value of the number variable) is NOT equal to the string "10" since, again, the while condition is ignored the first time. The do statement would still run at least once even if you had made the number variable equal to the string "10".
3. In the first execution of the do statement, assign the string response (return value) from the prompt to the number variable. So yes, the value of the variable is overwritten.
4. Now the while condition is checked. If the number variable, which is now equal to the response from running the do statement the first time, is not equal to "10" (resulting in a true statement), we run the do statement again. This keeps happening until the number variable (reassigned by the prompt response every time) is equal to "10" (resulting in a false statement), breaking out of the loop and triggering the alert.

Hope this didn't confuse you even more.