To start with the error codes when you go into test mode to get the code it only shows it the once then deletes itself. As to the 15 10 this isnt a code its the software version of the control board so thats nothing at all.
But E11 is a motor related problem. It can be either the motor itself, Worn carbon brushes in the motor, wiring to the motor or control board or it can be the control board itself at fault.
But as you say the drum is turning in test mode both ways and the ramps up into the spin then the motor is fine as it was the motor it wont do that. So its possibley the control board at fault as ths can play up like this and stop it from spinning during the wash cycle.
But a few checks to do first just in case is lean the washer backwards and get someone to hold for you.
Then lie down and look under the washer, There you will see the carbon brushes in the back of the motor. Remove these and see if they have worn. If so then you need to replace them. Also check the wiring plug to the motor as wires can break in there.
But if thats all ok then its looking like the control board is the issue here. As its not uncommon for these to fail as ive replaced loads of them in the past as they do fail alot on these Bekos.
So check them and see what you find. If you need any more info at all then just get back to me.