Gaining a level is a normal activated ability. It uses the stack and can be responded to.

You can't activate the first level ability of a Class unless that Class is level 1. Similarly, you can't activate the second level ability of a Class unless that Class is level 2.

Each Class has five abilities. The three in the major sections of its text box are class abilities. Class abilities can be static, activated, or triggered abilities. The other two are level abilities, one activated ability to advance the Class to level 2 and another to advance the Class to level 3.

Gaining a level won't remove abilities that a Class had at a previous level.

Each Class starts with only the first of its three class abilities. As the first level ability resolves, the Class becomes level 2 and gains the second class ability. As the second level ability resolves, the Class becomes level 3 and gains the third class ability.

There's no restriction on how many Class permanents you can control, whether they're the same or different classes. Each Class permanent tracks its own level separately.