I do not think so. I think the mechanisms of the enzymes would have specifically evolved according to the various selective pressures each organism has had to deal with in the past

I work in systems biology. We build computational models of cellular processes involving bio-transformations affected by enzymes.

In my research we focus on the effects of pH on the glycolytic enzymes in Escherichia coli and Saccharomyces cerevisiae. What I have seen so far is a definite difference in the way the first two enzymes, PGI and PFK, respond to pH changes in the two microorganisms.

I will attach a copy of my recent conference poster. This shows preliminary results of what I said above.

Do take into account however that this is raw data, and is not indicative of our final findings, which we yet have to make.

Poster AN INVESTIGATION INTO THE EFFECTS OF pH CHANGES ON PHOSPHOGL...

Sure , anything can happen in a Earth life context.

But generally speaking you  would not expect significant changes in action mechanism in a given enzyme in, let us say, different mammals. Even a given DNA polymerase in procaryotes has a similar action mechanisms than the equivalent enzyme in humans. 

In summary, for a given enzyme for a given substrate the action mechanism in a rodent or in human will be very similar.

The actual catalytic mechanism (and the accompanying active site sequence and structure) are often highly conserved across species. The regulation of the enzymes, and their role in signalling networks (see answer 1 above) is often not well conserved at all. 

So by that, I assume their kinetic equations would remain same?

In the absence of allosteric modulators (as Gautam points out, regulators or the mechanisms of action of regulators may differ),  yes

In agreement, as long as you are not considering allosteric modifiers, the equations we use across different species remain the same. When taking allosteric modifiers into account the equations would change according to the modifiers you are looking at. For example, if you are looking at PEP inhibition in terms of PGI, you would have to add an inhibitor term to your equations. I can provide you with the equations we use and the texts they are based on if you would like?

Here is an enzyme for which one might argue either way:

EC 4.3.1.9

See the Comment section of this KEGG enzyme entry

http://www.genome.jp/dbget-bin/www_bget?ec:4.3.1.19