Dead time is an application specific parameter, you won't find it in a XOR datasheet. Dead time is caused among other reasons by the NOT gate propagation delay inside the XOR circuit. Propagation delay spec in data sheet gives you a measure of the gate speed, it makes clear that standard 74HC86 is unsuitable for 2 ns measurement.
Because it's only part of the total propagation delay, you can assume that dead time is always smaller than specified propagation delay, may be e.g. 1 ns for 74LVC86A. In so far the device should work for your requirements. To measure smaller time differences around zero, you should absolutely follow KlausST's suggestion of applying an additional delay to one input signal.
You reported that in a certain range of input signal delay, the output signal doesn't change. This is probably due to ground bounce and insufficient power supply bypassing. Similar effects can be expected with all real setups, but depend much on the circuit layout.