What does the ‘−1’ superscript mean in units?
The -1 means "per" unit. So your first example mol/L-1/s-1 is not correct - it would actually be written as mol L-1 s-1, OR mol/(L s). It is also sometimes written as mol/L/s, but the double division is ambiguous and should be avoided unless parentheses are used.

If it were mol L-1 s-2, this would mean moles per litre per second per second.

This is really just a question of notation, and is not chemistry-specific at all. Yes, all the minus/plus signs and the value of numbers are important. Good examples of units can include:


area, measured in m2, or metres squared
volume, measured in m3, or metres cubed
pressure, measured in N m-2, or Newtons per metre squared
velocity, measured in m s-1, or metres per second
acceleration, measured in m s-2, or metres per second per second