私の英語が間違ってたら遠慮なく指摘してください。って英語でなんて言うの？

Verb phrase could tell asking person to perform that action.
Similar in third example can point out again asking for an action to be done

Either one of these phrases is appropriate to say and is a polite way of telling a teacher/listener to correct your mistakes. By using the word "interrupt" in the second example you are giving the person permission to stop you abruptly and correct a mistake that you have just made.

To slip up = to make a mistake or error when doing something.
To alert someone = to warn/nform someone about something they are not aware.
"I'd like to alert you to the fact that my son is only 15 years old."

"Please let me know if I make any mistakes." is both a polite and casual way to ask your teacher to tell you when you have made a mistake.

If you want to let the person know you really want to know something and wouldn't be offended you could say 'don't hesitate' this means you are happy to be corrected
'Could you please tell me' or 'Please let me know' these are both polite ways of asking them to do something for you

Native speakers will say like this: “Please, don't hesitate to correct me”.
There are lots of other ways to say it, but this one is the most accurate and most commonly used.
Here is an example on how to use this in a conversation:
A: Teacher, please, don't hesitate to correct me if I make mistakes.
B: Alright. So far there have been no mistakes!

If you are still unsure about your English speaking, it is okay to ask someone to point out any mistakes to you. If you let them know you are still learning and want to be corrected, then it will put the person at ease and make them feel comfortable correcting any mistakes you might make. Let them know that you appreciate them helping you learn and that you want to learn the correct way to speak English. You might even ask them if you are unsure, "Did I say that right?"