Height of the substrate in online PCB impedance calculators
1) Yes, H is the thickness of the prepreg between the top layer and the plane in layer 2. After pressing the board.

2) Yes, if you mirror the configuration from top to bottom the impedance will be the same.

3) Yes and No. The term is "plane" (not "ground plane"). The fields can't read... :-) Planes will work the same no matter what you name them.

Note 1: The impedance tolerance after manufacturing may be around +/-20% on outer layers and +/-10% on inner layers.

Note 2: This and other formula based impedance calculators may easily be way off if you have thin dialectric and skinny traces (which is what we often want in order to squeeze a lot of traces into a given area). If you want more accurate results in those cases, find a 2D field solver like the one in Hyperlynx og Cadence SigXplorer. There is a free one on the net called TNT as well.