In my experience, I would expect that the thickness variance across a 24" x 24" layout to be less than the width of the kerf.
If you are planning on using acrylic, I recommend using extruded acrylic (as opposed to cast) for designs in which the thickness consistency is very important. Extruded acrylic generally has more consistent thickness throughout the sheet due to the differences in the manufacturing process. However, manufacturers do not specify anything regarding the possible variance like you are describing, only the overall thickness tolerance of the sheet, and we would not be able to ensure or accurately check this before cutting your parts.
Depending on how serious your requirements are, you might consider looking for tight-tolerance sheets or having material machined down to an exact thickness and shipping it to us for cutting (after you request and receive a quote); we have had some customers do this in the past.
Regarding your concerns about cutting error, the general idea is that the laser cutters will center the laser beam on the lines in your file, so your parts should be what you specified to within the kerf.
I hope this information helps; please feel free to contact us if you have any additional questions.