Is it possible to run android without a screen on plain phone mainboard? I use Android phone as "brain" of my DIY robot in connection with Arduino Mega board. It is generally low-cost solution, because cheapest Android phone costs about $30 and Arduino Mega clone costs $7. So it is cheaper than most of single-board computers and have more i/o pins than all of them and have all needed possibilities: wifi, camera, GSM, GPS, gyroscope etc.
But I want to decrease price of the solution and decrease it's weight. So I could buy only mainboard of a smartphone without excess LCD screen, box and battery. And run it.

But I don't know, whether it is possible to run mainboard without a screen. Is it possible to run it and also reinstall Android via USB if I need it?

I know that most of single board computers allow to install Android. But I mean indeed a mainboard of ordinary phone.
If you can prepare the phone before removing the screen with Tasker (or similar tool) and plugins like Autoinput and test it well, I think you can obtain the results you are looking for.

Run and install android from USB it is going to be (if possible) a very tricky mess.

Nonetheless, if you are able to put the phone in download mode you could be able to work on this topic too.