-Suppose that the distance between "D1" and "D2" is too large.
-Does "D2" give a delayed response compared to "D1"?
In the case shown, with both diodes near the power supply but with a very long wire between them, there will be no delay. However, in the case shown below, there will be a delay:
The reason there's a delay in this case and not in the other is related to the fact that information must travel at the speed of light or slower, so the information that the power supply has been turned on has to travel through the Extremely Long Cable™ at no faster than the speed of light.
For further research, I recommend websearching the term "transmission line".
Yes, D1 will emitt light a very short time earlier than D2. The current flowing through D1 will have to load the the very long cable's capacity before the same current will flow through both LEDs.
No, for an ideal diode (or LED) there is no capacitance or inductance and the current through both is equal. (and in the ideal world wires are made out of superconductors)
For a real world diode there is a slight amount of parasitic capacitance and inductance that might cause a slight difference if you're looking at very small timescales, for most applications this would be negligible.
If you wanted to analyze this for a distance that is 'too far' then model it as a transmission line ( a long line of RC or RLC filters) or insert the calculated of resistance of the wire or trace between diodes between diodes