Internet Explorer 10 has so-called compatibility mode, which can be activated to make older sites display properly.
If the compatibility mode is enabled, does it slow the overall pages performance and/or JavaScript? And is there any other impact apart from performance eg. overall stability, memory usage etc?
After a bit of thought, I think the answer to your questions is:
It depends.
It depends on what your site does with css. It depends what your site does with markup. It's entirely possible that some new CSS thing has come out in the last few years that IE 11 added very good support for, and the team of engineers spend lots of time optimizing. The old legacy way to do it might work, but be slower because it's old code that never got refactored and optimized. Imagine that repeated over and over... an old IE8 feature that now has a much faster IE11 css or markup equivalent... and you're not getting to leverage any of it.
Or, maybe your site uses none of that functionality, and it's (about) the same no matter which mode you're in. While the likely scenario is that you will probably see some sort of performance hit by running in compatibility mode, it certainly isn't guaranteed.