Store the protein samples frozen at -80 C or in liquid nitrogen. You can include 10-20% glycerol if you think it will aid stability. When you do the ATPase assay, the reaction will probably be slowed down somewhat by the glycerol if the sample is not diluted, but it should not otherwise be a problem.

In my experience glycerol does not affect activity, but protects from frozen. You can either add glycerol up to 30%w/w and store your protein at -20 taking aliqoutes when you need, or use glycerol-free buffer for the storage, in this case you should prepare several aliquotes, freeze them and use one probe per one experiment, without freeze - unfreeze cycle.

To know for sure at what final glycerol concentration it starts affecting the results, do now a parallel assay without and with different concentration of glycerol. This also gives you a reference value for the specific activity of your prep to compare your results after storage to.

It certainly depends on the system. I was studying a GTPase once and small amounts of glycerol definitely slowed down the reacion.

All osmolytes bind water, which is needed as a "grease" to aid the conformational changes during enzyme turnover. Therefore, all osmolytes reduce kcat. With some enzymes this effect is stronger than with others. Just make sure that the osmolyte concentration is constant over all your experiments.

Glycerol/water mixtures have much lower freezing points than water (see doi:10.1021/ie50189a017 for a phase diagram), you can therefore store enzymes in the liquid state at -20 °C and thereby avoid freeze/thaw cycles.