Regarding the search inputs, they are currently structured within a form. When you update the results, even through ajax, the entire form gets updated. This also leads to the replacement of the current form along with the View results.
I'm afraid achieving what you desire using Views alone might not be feasible. A custom solution is necessary for ajax submissions after each keystroke/keypress + handling the returned matching posts to populate the results.
I'm afraid that this is beyond the scope of our support policy.
However, I can suggest you some helpful tutorials that cover using ajax in the context of WordPress, such as the one available at: hidden link