All experience counters are identical, no matter how you got them. For example, the last ability will count experience counters that you got from the second ability, from another ability, from another copy of Azula, Ruthless Firebender, and so on.

The experience counter goes on you, the player, not on Azula. You will keep that counter even if Azula, Ruthless Firebender dies.

"Firebending N" is a keyword that represents the ability "Whenever this creature attacks, add N {R}. Until end of combat, you don't lose this mana as steps end."

Any other mana you add during combat will still be lost as normal when moving between steps of combat.

Firebending abilities aren't mana abilities. They use the stack and can be responded to.

Mana from firebending abilities isn't lost until you leave combat and go to your second main phase. It can be used at any time during combat, even after combat damage has been dealt.

Multiple instances of firebending on the same creature trigger separately, each granting you the appropriate amount of mana.