How telerik 2013.1.806.0 support the future versions of chrome?

Unless Google introduces some breaking change in the Google Chrome browser Test Studio will be compatible with all future versions of that browser. This means, in most cases, there's nothing you will need to do to run your tests in the new browser version.

In the event that Google does introduce a breaking change we work very hard on changing Test Studio to be compatible with that change. We do our best (though we cannot promise) to have a release within two weeks of a breaking change like this. That is our policy for all browsers that we support.

Yes, it equally applies to recording in all browsers.