Can a Z84C00 CPU directly drive 74HCxxx series logic?
I'm also not sure what the subscript "1" and "2" in the symbol name
  here is used for


Look at the end of each line - #1 is for a loading current of 1.6 mA and #2 is for a loading current of 0.25 mA. This means that if your 74HC chips don't take much current (you'll have to check) then you are probably going to be OK with a supply at 5.5 volts.

The input currents to HC series are much less that 250 uA but you will need to confirm this in the data sheets of the parts you intend to use.