Programming/Debugging with ICD3 and PIC16F819-I/P
A1. MPLAB will set the debug bit as necessary.  Note that in MPLAB there is a Debugger menu as well as a Programmer menu.  Which one you use determines how it sets up the PIC.  A PIC programmed from Debug may not run if the ICD3 is removed.  To run without the ICD3, program it from the Programmer menu.

A2. The ICD3 does not provide power to your device, so you need to apply power externally.  LVP doesn't usually have to be connected.  There have been a few parts that needed this line held disabled in order to be recognized, but I haven't seen that in a long time.  MCLR will operate properly from the debugger, but you will need at least a pull up resistor there when you attempt to run without the ICD3 connected.

A3. The PIC should be recognized by the ICD3 as soon as it's been selected as your Debugger.  A message appears in the output window as to whether it sees it or not.  Note that if you are using the internal oscillator (thereby not having to add an external crystal), your program should specify that the internal oscillator is to be used.  Otherwise, you may be able to program the PIC, but it won't respond afterward.  You don't "lose" the part, but you do have to fix the error and program it again.

A4. You can do all those things you described from inside MPLAB.  I hesitate to use the word "simple", as you have to get everything set just right (example: choose the right chip from "Configure") before it works.  After the "hello world" exercise it becomes easy (not counting writing the software, of course).