Could you please check these few sentences, please? Thank you very much!

1) Can't you say "cheat in the maths test"or is only on possible?
She invites me for a walk (is it a mistake? She invited me to go for a walk)
2) She makes a living by teaching.
She was sorry (is it possible instead of "she regretted"?) because I hadn't read that book.
3) It is reported that she lived in London. (Active: Newspapers report that she lived....)
4) She recommended me to take an umbrella (corrections: She recommended that I should take an umbrella/that I take an umbrella)

1. "on" is the word used in the US. I'm not sure if it's different in the UK.

She invites me for a walk. ~~>unusual phrasing
She invited me to go for a walk. ~~>standard phrasing

2. She makes a living by teaching. = OK

She was sorry that I hadn't read that book.
[To regret something is to be sorry for something SHE (or whoever the subject of the sentence is) did.]

3) It is reported that she lived in London. (Active: Newspapers report that she lived....) = OK

4) She recommended that I should take an umbrella/that I take an umbrella.
(If you say "she recommended me..." then you're saying that she is praising you; she is saying you're the best for the job.)