Do you wash it before every single use? Every time you put a fork down on it, or maybe a piece of fruit that you just washed in the sink? Or maybe you ordered some take out and didn't cook anything, but you still touched the counter after washing your hands?
If you're really that careful about cleanliness then sure, but I'd guess the vast majority of people are not.
Also if the cat knows it's allowed on the counter, it's probably not going to care if you just washed it or are actively in the process of cooking.
I'd be more concerned about the abrasive factor of dishwasher soap over time, rather than temperature issues. Dishwasher soap is very gritty since it needs to use that as an abrasive to get food off, rather than someone wiping with a scrub.
Really gonna depend on what type of print sheet as well. If you have a coated sheet, that coating is probably going to get worn off.
Often it's distributed databases like Google spanner, amazon aurora, cockroach db, etc. Google has a public whitepaper on spanner you can read.
https://static.googleusercontent.com/media/research.google.com/en//archive/spanner-osdi2012.pdf