With a wireless charger circuit (ignoring the 7805 voltage regulator part) your receiver coil is trying to pick up the alternating magnetic field from the transmitter. It can have a good coupling or a less-good coupling as per the picture in one of the links: -
On the right with the receive coil further away it receives less magnetic flux. So if you looked at the voltage on your receive coil you would see an AC voltage that got bigger as you approached the transmit coil.
You could do a different test that has the transmit and receive coil fixed at a certain distance. That test would be to load the receive coil with a resistor and you would find that as you try and draw power from the receive coil, the receive coil voltage would fall.
Hanging a diode bridge and smoothing capacitor on the output makes very little difference except you'd be looking at a dc voltage instead of an ac voltage.
So that's the backdrop and of course the 7805 regulator and its output load want to consume a certain amount of power - that power is determined by the load connected to the output of the 7805 and the 7805 output voltage (5V).
When I disconnect my regulator(leave the circuit open) I see my DC
output like 30-40V, when I close the circuit (connect my regulator) I
see voltage between 7 - 20V
Hopefully that should be clear now from the explanation above.
Does the resistor (load) connected to regulator have any impact on
this decrease in voltage?
On its own the 7805 will consume a few tens of milli watts but when the load resistor is connected to the 7805, that power will increase because it's using power to charge a battery (battery voltage x charging current). This might mean 5V at 1A = 5 watts but there were no details in the question about this.
Is there any relation relating the load resistance and input voltage?
Or any relation for impedance matching between them?
Yes, load resistance changing will affect the dc voltage level because even when the receive and transmit coils are adjacent there will be an imperfect leaky transformer coupling and voltage will lower when more watts are required to charge your battery.
For impedance matching, you could make an argument for developing a circuit that tried to optimize the max power output from the coils. This would toss-away the 7805 voltage regulator and replace it with a buck-boost regulator and power monitoring circuit (the same as what is used in sophisticated solar panel charging circuits to optimze the power transfer based on amount of sunlight). But this is way too complex for a simple non-contact low power charging circuit.
To the resistor(load) to receive maximum power are there any design
calculations to be done for regulator?
I've designed systems like this to provide magnetically coupled power to electronics on rotating machines and the calculations rapidly become too cumbersome so I simulate the leaky-transformer effect of the transmit and receive coils. If a working prototype isn't good enough (say for transferring a couple of watts at 40mm gap I rethink, rewind and retune.