Is Neural Network Architecture independent of Data?
Because your two questions are phrased a bit differently, I'm going to answer the one in the title.


  Is Neural Network Architecture independent of Data?


Generally speaking, yes.

One simple reason why you might have to change the architecture is if your new task has a different number of classes. If so you must change the last layer of your network. 

Another reason is that some deep CNNs perform multiple stages of spatial subsampling (e.g. through pooling). This requires the input to be larger than a minimum shape. E.g. InceptionV3 and InceptionResnetV2 require the input to be at least $75 \times 75$, Xception $71 \times 71$, and so on.

Even if you don't necessarily have to, sometimes you should. An architecture that does well in MNIST, won't necessarily do well on ImageNet, because the latter is much harder. It requires models with a higher capacity.