Suppose at the instant a gravitational wave passes through an interferometer, one of the interferometer's arm get stretched by 1%. Would the wavelength of the photon travelling in the arm also get stretched by 1%? If so, then would the frequency of the photon remain the same and hence increasing the velocity of the photon? Could the velocity of the photon be >c, just like how distant galaxies can move faster than c due to the expansion/stretching of space? If the velocity of the photon is higher, then why would it take a longer time to travel down the arm when space is stretched?

It's mentioned here @5:50 that the wavelength get stretched:

Gravitational waves oscillate. The same wave will lengthen then shorten the length of the arm of the interferometer. There's no net change in distance.

It's mentioned in the video @6:00 that the oscillations created by gravitational wave are very slow compared to the time it takes for a photon to travel down the arm, and so there is a net change in the distance travelled by that photon.

1. That's because the gravity wave has a very long wavelength. The gravity wave itself still travels at the speed of light. It's just that its wavelength is much longer than the arm of the interferometer (the interferometer can't measure gravity waves with shorter wavelengths).
2. The distance has changed, but the speed of light has not. So yes, the photons that the interferometer bounces back and forth take a big more time to do a round-trip when the gravity wave has lengthened the arm, a bit less time when the gravity wave has shortened the arm.
3. For distances much larger than the gravity wave's wavelength, there are many oscillations so there's no net change in distance.

Why doesn't the speed increase when the wavelength is stretched by the expansion of space? Like how the receding speeds of distant galaxies increase with the expansion of space.

Because you're using different notions of "speed" in each case.

The speed of light is a local speed. It's how fast the photon is moving past stuff local to it. Photons always travel at speed c in a vacuum locally, no matter who is doing the observing. From this definition of speed, photons traveling the interferometer arm at different times have the same speed, but take different amounts of time depending upon the gravitational wave state.

The speed of a far-away galaxy is different. That's a relative speed of two far-away objects (the observer and the far-away galaxy). General Relativity has no firm definition for what that kind of speed even means. There are multiple possible ways to write down such a speed, and they will typically disagree with one another, and nobody can say that one definition is any better than another. This is why recession velocities can be faster than the speed of light, and why far-away photons traveling towards us can get further away.

Note that if the length of the interferometer arm changes appreciably as the photon crosses the arm, then it will likely pick up a redshift or a blueshift, as the length change will also affect the photon's wavelength.