With your old pads, it sounds like the pedal motion was already excessive (5cm pedal motion is too much with four wheel disc brakes, about what you'd expect with rear drums in need of some adjustment), so you may need to bleed them.
With new pads the pedal can feel soft until the new pads bed in to the rotor surfaces, i.e. the new pads will be nice and flat, but the rotor surface may not be, and the pad material that is touching the high spots on the rotor will compress as the piston pushes against them, then decompress as the pressure is released (pushing the pads away from the rotor in the process).
The problem is worse with rotors that are worn with grooves etc, but gets better as the pad wears into the uneven rotor surface and the piston pressure can then be applied over the entire pad area rather than just a small % of it. When this problem exists, the brakes are far less effective in hard useage until the pads bed in, so be careful.
Your pedal going all the way to the floor after you fitted new pads may be a combination of gas in the hydraulics, and un-bedded pads.

 
			
 
			
			 
					
					
					
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