This is a difficult question to answer, in that people have long argued about the precise reaction mechanisms of well-characterized enzymes. However...

For part (a): you could start by looking at pH dependence of reaction rate, differential reaction rates in H2O and D2O, effect of mutations, 'fast' enzyme kinetics measurements, to look for burst phases and product inhibition, etc. Whole books on this topic (e.g. Fersht)

For part (b): almost certainly not, in that an unbiased approach would need QM/MM-type calculations which are very compute-intensive and not easily extensible to enzymes - for a starter you would need an accurate 3D structure of the substrate(s)-bound state. 

The literature on related enzymes should be consulted for likely reaction mechanisms.

As far as I know, there is no computational tool to predict the reaction mechanism,

To find which enzyme residues are important, usually we can see by blast your protein sequence, and align your enzymes sequence with enzymes related,

For example, your enzyme is shikimate kinase from A organism, so you can check about shikimate kinase from B organism in other paper.

after that you can see the conserved residue, usually conserved residue are important residue in the enzymes. Check about conserved residues role in other paper.

You will see if there is a paper related with your enzymes.

If you know the substrate and product, from the chemical structures, you can likely determine what type of enzyme is performing the catalysis, because in biochemical systems, there are a limited number of chemical mechanisms. Then once you know what reaction takes place, you can compare to other enzymes in the same class to predict what residues perform the catalysis.