Why can't the birdcage be cut? Why does it appear that Doflamingo's Bird Cage is indestructible? What is the different between the strings used in DD's attacks and the birdcage? Why can't Zoro cut the birdcage?
I don't think it is because Zoro cannot cut the cage that he is trying to push it back. In chapter 270 he was in a similar situation, where he was trapped by the White Cage during the Ordeal of Iron. At that time he says it is completely unnecessary to trap him, because he is not going to run away. Now, with the birdcage, the situation is similar in the sense that he has no intention of running away. Cutting the birdcage or running away, will not benefit anyone with Doflamingo still around. He will try to kill anyone present at the island, with or without the birdcage.

Zoro knows this and knows that the person that will defeat Doflamingo is Luffy, but he does not know how much more time Luffy will need. Therefore to buy Luffy as much time as he can and to prevent as much casualties as he can, he tries to block the cage, instead of cutting it. We have no clue what would happen if he cuts the cage and it will probably require much needed energy, which will probably just go to waste by cutting the cage over and over again, because we all know Doflamingo will keep putting it up again and again.

Now as to your question whether the birdcage is indestructible, that has not yet been revealed up until the latest chapter 787. Other than a few unknown pirates and the meteor from Fujitora, there has not been a single person that has actually tried to attack the cage, not now, nor in the flash-back. Therefore it is hard to say how strong the strings actually are.