But, without performing any sort of testing, is there a way to estimate the value of the inrush current, provided we have the capacitance values? Or only after testing, do we get to know the inrush current value for the circuit?
Yes, it is possible. However, as the complexity of a circuit increases, it becomes harder and harder to estimate it, especially if you are using some current controlling mechanism, whose behavior is not linear.
Assume the following simple RC circuit, where the current in the time domain is given by:
At \$t=0\$ you would have you peak current, which equals \$10A\$. Determining such a equation for more complex circuits can be challenging, and the traces inductance also places a role in limiting the peak current. Furthermore, there is a dependence on the supply voltage's slew rate, which can shape the inrush peak current.
I suppose the best way of estimating the in-rush current is by simulating it an considering the biggest capacitances in your circuit. It gives you a good start and you can better adjust the current limit later.
simulate this circuit – Schematic created using CircuitLab
Suppose, we have figured out the inrush current. To limit the magnitude of the inrush current, we may employ a small value of resistor so as to not dissipate high power in the resistor during the standard operating conditions. But how to calculate the peak power dissipation during the inrush condition?
You way of thinking is correct, and the inrush current can be limited using a NTC in series with the supply voltage. Once the system is starting up, the NTC poses a large resistance to the supply, but as it gets warmer, its resistance drops considerably. Basically you limiting the current at start-up and reducing the power loss at steady-state. However, using NTCs has catch, if your device is being switched on/off very often, the NTC cannot cool down, thus, not being as effective as it should be.
Is it simply ((I^2 * R) * 5ms)? Or what is the formula for calculating the inrush power dissipation?
You are being a bit conservative with your formula, because it assumes that your peak current lasts 5ms, which is not the case. Usually the in-rush current has a bell shape and for the precise value, you would have to integrate the area below it. That said, your formula should be ok, if you want to be in the safe side.
Here there is an application note by Texas Instrument, where many approaches for limiting the inrush-current are presented.