I watched a short introduction video online. There are few concepts which are a bit confusing.

1. The video said the laser is single mode (monochromatic), the wavelength is 780nm, so what is that wavelength really refer to? What I am asking is what's the wavelength related to the medium creating that laser? Is that the atom level of the medium generating that laser (sorry, I don't have much background on that)

2. In some book, it said the profile of the laser is Gaussian, but when laser is in transmission, what's the wave look like? Is it sinusoidal wave of the form $y = \sin(kx)$ where $k=2\pi/\lambda$, if so, why they said it is Gaussian?

3. I the undergraduate course of physics, I remember it said the plane wave have the form $\sin(kx - \omega t)$, so the speed of the wave is $v=\omega/k$. But if the laser wave has the form $y=\sin(kx)$ and transmit in the air, why there is no frequency term? So the speed is ZERO?