1. Tickets arent sold for kids below 5 on the Indian Railways - you do have to enter the child's name into the booking form, or the website

2. 2A / 3A etc on Delhi to Agra trains are more likely than not on long distance trains. For indian railways, I'd make sure that you're on a point to point train or one that starts from Delhi rather than where Delhi is an intermediate stop.

And of course pick whatever time works best for you. A 4 hour trip is nothing on indian trains so the class doesnt matter.