Would there be no time in a universe with only light?
A universe containing only light is simply a radiation dominated FLRW universe. Indeed our universe had approximately this geometry in its radiation dominated era. The FLRW metric is a perfectly good spacetime, so time certainly exists. Moreover the geometry is time dependent so we can use the energy density as a measure of time.
I concede that it's hard to build a device capable of measuring time in a universe containing only light, but to claim that time does not exist in such a universe would be plain wrong.
Speed has a definite meaning, i.e. , the absolute value of the velocity vector, which is given as $({\rm d}x/{\rm d}t,~{\rm d}y/{\rm d}t,~{\rm d}z/{\rm d}t).$ There is no mathematical meaning in the phrase "you travel through time at the speed of light".
Does that mean, if our universe would be filled with light only, no time would exist?
No. Entropy would still be defined by its statistical definition, and an arrow of time would exist.
Is the existence of mass therefore necessary for the existence of time?
Not of massive particles. As long as there is energy in its general relativistic meaning
$$m_0^2c^2 = \left(\frac{E}{c}\right)^2 - ||\mathbf p||^2$$ in natural units, where $c =1,$ $$m_0^2 = E^2 - ||\mathbf p||^2\,.$$
there will be an invariant mass of the photons comprising the light. In addition recent studies show that an arrow of time exists just because of gravitational interactions .
I think you are becoming confused with the notion of proper time. Proper time is a measure of how much passage of time is experienced by an object. When you say that an object sitting still travels through time at the speed of light, you are kind of right. Let me start by explaining this.
"Speed through time"
If we're not experiencing any forces (gravity does not count) due to other objects pushing us or somesuch, then we may consider ourselves as being still and everything else as moving. This is called a choice of reference frame. In picking this reference frame, we have our own measurements of spacial coordinates $x,y,z$ and of how much time has passed-- i.e. by looking at our wristwatch. We call these coordinates $(x,y,z,t)$.
An object's trajectory through spacetime can be represented by a velocity in this choice of coordinates. Without going into too much detail, the most natural way to express this velocity is by how far through space and how far through our time coordinate an object is travelling per second of time that it experiences. Remember, this is called proper time and we denote it $\tau$. This is how we would express an object's velocity through spacetime:
$$\left( \frac{dx}{d\tau}, \frac{dy}{d\tau}, \frac{dz}{d\tau},\frac{dt}{d\tau} \right)$$
Again without going into too much detail or technicalities (I can if you want me to, but I suspect that you probably don't have the necessary background), special relativity tells us the amount of time we have to wait for an object to 'experience' one second ($dt/d\tau$) as a formula of its speed as measured by us ($dx/dt, dy/dt, dz/dt$). This factor is often denoted $\gamma$.
$$\gamma = \frac{dt}{d\tau} = \frac{1}{\sqrt{1 - \left(\frac v c \right)^2}} \quad \text{ where } \quad v = \sqrt{\frac{dx}{dt}^2 + \frac{dy}{dt}^2 + \frac{dz}{dt}^2}, \quad c = \text{speed of light}$$
What you call an object's 'speed through time' may be thought of as the amount of proper time this object experiences per second of our own time. In other words, how many minutes pass on a moving stopwatch per minute that passes on our own? This quantity is $d\tau/dt = 1/\gamma$.
$$\frac 1 \gamma = \sqrt{1 - \left(\frac v c \right)^2}$$ This is actually the equation of a circle! $$\left(\frac 1 \gamma \right)^2 + \left(\frac v c \right)^2 = 1$$
So an object's 'speed through time' and speed through space (as measured by us) always lies on the edge of this circle:
Would time stop without mass?
As you can see, an object can either be moving quickly through space (again, according to us), or quickly through time, but not both. A photon has no mass and a consequence of this is that it is always moving at the speed of light. As a result, its arrow tip in the circle above is always pointing straight up, and so it experiences no passage of time; in other words, no proper time. (An interesting note as an aside here is that if the photon were theoretically an unstable particle which decayed, we could never observe the decay because time never passes for the photon!)
So what you are asking is whether time would 'stop' if the universe was all photons. While true that time would never pass for any of these photons, there are still some aspects of the universe's spacetime which we do not understand, and which depend on time. For instance, the universe is expanding. If the universe were all photons, then this could still happen, and would involve something changing about the universe as the time coordinate advanced.
So without any mass in the universe, there would be no passage of proper time (i.e. watches would never tick; biological processes would freeze) but there would still be a notion of time because the universe itself is changing with it.