Will excess current damage my ICS?
The ICs will draw whatever current they want to (if the power supply can deliver it). The regulator just needs to be able to deliver the maximum current your circuit wants to draw. 

Some 'headroom' is necessary to ensure that the circuit won't be starved of current, but too much makes it more vulnerable to damage if overloaded (eg. due to incorrect wiring or a short circuit). Therefore you should choose a regulator that can supply more than the expected peak current draw, but not a lot more. 

For experimenting I recommend using an adjustable current limiting power supply, or at the very least an ammeter in series with the supply. If that is not possible then use a regulator with a lower current limit (eg. 78L05 or 78M05) and check its temperature during operation of the circuit.     

Finding the maximum current draw of a particular IC can be a bit tricky. You can add up IOL of each output, but that may exceed the maximum current or power dissipation rating of the entire IC. The IC will draw some internal current even when the outputs are disconnected, and this internal current may go up or down in different logic states. Often the only way to find the actual current draw is to put the IC in a circuit and measure it.