It sounds like you actually pretty much get it. Extra Attack stacks with any other abilities that give multiple attacks, like double. So say a character has Double Attack and is granted Extra Attack; they can now make 3 attacks (note that you dot actually gain the "Triple Attack" ability, although the difference between Triple and Double+Extra is mostly semantics). Likewise, if you have Triple Attack and are granted Extra, you can now make 4 attacks (but still don't have the ability "Quadruple Attack", but again, that's mostly semantics).

Now say a character has Double, and a commander grants them Double again. In that situation, they still just have Double, so they only make 2 attacks. If they had Triple and were granted Double, they still only get to make 3 attacks. That's why a CE granting Extra Attack is stronger than one granting Double Attack: if you already have something giving you multiple attacks (like Double or Triple), Extra adds another attack. Double does not. (Extra also stacks with itself, so if you have Double ad 2 different commanders are giving you Extra, you now have 4 attacks)

As for Twin, yes, you get a Twin attack on the Extra attack, because it says "Whenever". If you have Twin you get a second attack EVERY time you make an attack, whether it is a Double, an Extra, or even an attack granted by a CE.