Hello and welcome to iMore, to answer your questions....
1) The only ways to mute notifications are with the silent switch as you mention (turns off sound and vibrations if you have it set that way) but they still show on the phone, second way to go to settings/notifications and adjust the options to best suit.
2) Apps remain open in the background, allowing faster movement between apps etc, swiping up closes them (good for saving battery), the iheart app loads as the iPhone has more than likely learnt from pass use that when in your car and iPhone connected that is the app you want open.
3) Best way to search for items on your phone is pulling down from top of screen and using the search bar
4) No idea on aftermarket keyboards as I have always used the standard one.
Hope this helps feel free to ask anything else!
11-25-2015 04:53 PM