Hello there. Here the answers:
1) The positive option is the Grey Button (I also consider that this is confusing, but you get used to)
2) The player will be able to train the main ability of the trainer. For Example, if my trainer gets me a bonus of 15% in Special Skills and 10% in Rally Skills, my player will be able to train both of this skills, as also Service and fitness.
You could hire an Asistent Coach to train all the Skills, but if his Skills aren´t good enough your player will loose some motivation.
3) When you choose a Trainer to travel with a player, Automaticly will turn off from the other player.
4)The Player pays for the Trainer. You can choose wich one of your players will pay him. I´m not sure if there´s a way to make yourself to pay.
I hope this was helpful, sorry for the horrible grammar, and enjoy this awesome game!