What is the significance of i^Q mod P = 1 and are there any special properties to be aware of?
A special case of the relation you've found is called Fermat's little theorem.
In a more "mathy" way this looks like:

$i^Q\equiv 1 \pmod p$

which will always hold for $Q=p-1$ and for any $Q|p-1$.

The use of this theorem is limited in cryptography, however the problem of determining $Q$, if only $i$ and $p$ are known, this problem is called "discrete logarithm problem" (assuming there isn't a "1" on the right side but rather some random number).

If $p=r*s$ and you're given $Q$ then this is called the RSA-problem (the problem if finding the Q-th root modulo p), again assuming the right side isn't 1 but rather some random number.