IPad – Continue screen recording even after screen turns off

Try using Background Music.

When recording with Quicktime click the arrow and change the microphone to Background Music.

You can record system audio with Background Music. With Background Music running, launch QuickTime Player and select File > New Audio Recording (or New Screen Recording, New Movie Recording). Then click the dropdown menu (⌄) next to the record button and select Background Music as the input device.

You can record system audio and a microphone together by creating an aggregate device that combines your input device (usually Built-in Input) with the Background Music device. You can create the aggregate device using the Audio MIDI Setup utility under /Applications/Utilities.

Edit: If you have homebrew installed you can install using brew cask install background-music

^{source:https://github.com/kyleneideck/BackgroundMusic/blob/master/README.md}

The fourth point about rewinding video and re-recording is unheard of. Let's say you made a minor mistake while speaking, best you can do is one or more of the following:

• Open the video editor, add a text box that shows the correct thing to say, make it visible in the video as an overlapping window for 3-4 seconds. This is a common method to fix mistakes in post-shoot editing.

• When you realise that you've made a mistake, immediately repeat the last sentence with the correct word and let the recording go on. While editing it later on, trim the portion where you misspoke and delete that part. There will be an slightly abrupt cut but not a big deal in lectures.

All this is possible in iMovie for macOS. It's free and easy to use. Also use help centre: https://help.apple.com/imovie/mac/10.1/#/mov5ec96da08

For screen recording, Quicktime is a good option. If you want to appear speaking in the video, use Photo Booth and put it in a corner such that it appears on the screen.

Other way is to use a camera and use clips from both the camera and the screen in the final video.