As features in your input vector are of different nature, you should use different normalization algorithms for every feature. Network should be feeded by uniformed data on every input for better performance.
As you wrote that some data is skewed, I suppose you can run some algoritm to "normalize" it. If applying logarithm does not work, perhaps other functions and methods such as rank transforms can be tried out.
If the small decimal values do entirely occur in a specific feature, then just normalize it in specific way, so that they get transformed into your work range: either [0, 1] or [-1, +1] I suppose.
If some inputs have many zeros, consider removing them from main neural network, and create additional neural network which will operate on vectors with non-zeroed features. Alternatively, you may try to run Principal Component Analysis (for example, via Autoassociative memory network with structure N-M-N, M < N) to reduce input space dimension and so eliminate zeroed components (they will be actually taken into account in the new combined inputs somehow). BTW, new M inputs will be automatically normalized. Then you can pass new vectors to your actual worker neural network.