First note that any guest whose number is divisible by $3$ does not contribute to the lighting. Hence the light in room $3^kn$ will be the same as the light in room $n$ for all $k, n \in \mathbb{N}$.

Due to the cat's behaviour,
- $n \equiv 1 \mod 3$ turns any light red
- $n \equiv 2 \mod 3$ switches red and blue

Thus rooms with blue lights will be precisely the rooms where there are an odd number of divisors congruent to $2$ greater than its greatest divisor congruent to $1$.

To answer your first question, consider all numbers $n \equiv 2 \mod 6$. Its largest divisors are $n / 2 \equiv 1 \mod 3$ and $n \equiv 2 \mod 3$. Thus the light always ends up blue. Therefore the fraction of blue lights is bounded below by $1 / 6$.

More results can be found by considering different congruence classes of products of small primes. For example, if $n \equiv 1 \mod 3$ then the light is red; if $n \equiv 5 \mod 30$ then the light is blue. This gives some intuition for the linear behaviour. However more advanced techniques would be necessary to prove convergence.

Finally, neither the guests and the cat make their own decisions and rather follow a predetermined strategy. Therefore it is not really a game theory question and number theory is indeed the way to go.