Are there any home allergy testing kits that you can buy that work just as good as allergy testing at the doctor's office?

Yes, But... There are home allergy tests, but it really is safer to do it in the presence of a specialist, in case of an emergency. This is an interesting article regarding this.
No. You can have self-directed blood testing for allergy. This may be very helpful but depending on what you are testing for may be less effective than a "skin testing" for allergy. You should seek the advice of a board certified allergist to be sure that the test results are interpreted correctly, whether done by blood test or skin test. Interpretation requires context!