How to properly read Watt of a bulb power?
Why does a 100-w bulb consume 1 Wh in 1/100 hour, instead of 100 Wh in 1 hour? 


You are correct that 100 W for 1 hour is 100 Wh. If you divide by 100 you find that 1 Wh will be consumed in 1/100 h. If your energy meter LED blinks once per watt-hour then it should blink every 36 s if your 100 W lamp is the only load.


  Is 60-Wh equivalent to 60 * 3600 watt in a second?


We don't say "watts in a second" as the watt is power which doesn't have a time factor. Energy does, however, and energy is power by time so we can have Ws (joules / J) or Wh or kWh (common for energy billing).

Your maths is correct but your units are wrong. 60 Wh is equal to 60 * 3600 Ws (watt-seconds).

Note that SI units named after a person have their symbols capitalised but are lowercase when spelled out. 'A' for ampere, 'V' for volt, 'W' for watt, 'K' for kelvin (as opposed to 'k' for kilo), etc.