Using series of LM 78XX for power supply
Is using that strategy (shown in first image) good?


It's fine, and using multiple linear regulators will spread the power dissipation from the current across multiple parts. The problem is the current is in series, so the power dissipation will increase for the 7812 and 7809. The most important thing is to keep the regulators from going over their temperature rating. This means you must calculate the max current through each of the parts and multiply it with the voltage drop to find max power dissipated. The second thing is not to use more current than the part can tolerate. I would estimate the power of the LCD and ATmega to be under 200mA, but this needs to be calculated by you or measured, most regulators support at least 500mA and some 1A (depends on the package usually, which is not listed).


  As depicted in first image, input voltage is 24V. I Also use it for
  high power LEDs (in second image). Is there any side effect? (for
  example, voltage drops when LEDs are pulsing).


It is fine to use 24V for the LED's and for the regulators. One problem is the first regulator has a 12V drop, at 500mA (which is a ballpark number) this would be 6W dissipated. At 200mA it would be 2.4W and 100mA would be 1.2W. Make sure the 7812 has the right package and calculate the temperature rise