The tests are sat in mid-September, sometime around the 15th/16th usually. You don't have to apply for the school at the same time, this is done later on.
To give you a brief timeline...
May/June 2010 submit your application to sit 11+
Mid September 2010 sit exams
Mid October 2010 get results
Early November 2010 apply for schools
1 March 2011 get told your allocated school
Kent is an opt in system so you need to apply to sit the 11+ in May/June of Year 5 for your child to sit in Year 6. You can apply online at the Kent.gov website which is easy, and also means you get your results by email rather than waiting for the post.
Results are sent out about a month after sitting the exams - this year it was 19th October by email, and 20th October by post.
You now get the results prior to having to make the school applications (CAF) - Kent CAF deadline is today (6 November).
So yes your DS could stay at his present school in Kent and sit the tests there. The only thing I would caution is that you only get a couple of weeks from getting the results to submitting the CAF so this might not give you time to move house prior to applying. I'm not sure what happens if you move house closer to the school between submitting the form and allocations which aren't told to the parents until 1 March. I would call the Admissions Office at KCC to check.
And finally... (sorry, this has turned into an essay) if your DS scores very highly there are always Judd and Skinners schools which take the top scorers (no catchment area). Last year the cut off mark for Judd was 412 out of 420 and for Skinners was 406 out of 420, so it is competitive, but it doesn't matter where you live.