Did time pass more slowly in the past? 13 billion years ago, the universe was about 600 million years old according to many scientists. At that time, all matter in the universe would have to be closer, or in other words denser. 

Did time pass more slowly, at that time, relative to now? I ask, because increased gravity causes time to pass more slowly.

If we'd been there, and measured the speed of light as we measure it today, would the speed of light be different?

Finally, does distant stars look magnified, given that a "tiny" universe is stretched on our entire sky?

I assume then that we'd measure time based on caesium 133 and distance not as a function of the speed of light, but as a theoretical physical "device" that would be the same physical length then and now.
Without some reference to compare to, time passing more slowly makes no sense at all. But every clock has its own proper time that measures time along its own worldline. 

In cosmological models, the cosmological time is the proper time of a certain kind of ideal observer: one comoving with the Hubble flow. In other words, imagine space filled with observers for whom the universe looks as close to the same in every direction as possible (zero CMBR dipole anisotropy). The cosmological time is what's measured by their clocks.

As such, there is no time dilation... because we defined our time coordinate as that which is measured by the comoving clocks no matter how small or dense the universe is. You can see this in the standard form of the FLRW metric:
$$\mathrm{d}s^2 = -\mathrm{d}t^2 + a^2(t)\left[\frac{\mathrm{d}r^2}{1-kr^2} + r^2\,\mathrm{d}\Omega^2\right]\text{,}$$
where nothing happens to the time component no matter what the scale factor $a(t)$ is.

How could we compare the rate of clocks now with rate of clocks then? We could have the past clock emit some pattern, say a light at a specific frequency, and then later catch and measure it. If we find a different frequency, we could interpret this as evidence of time dilation. But we could also fix the clocks as our time standard and interpret the frequency shift as the result of a changing scale factor.

We could also arbitrarily rescale the time coordinate any way we like, but there's no point in doing so because it wouldn't change the time experienced by comoving clocks.




  are you saying that we've decided to look at it as an increasing scale factor where clocks are fixed, instead of time dilation - but in fact both are just different point of views?


That was phrased too poorly, so let me clarify. If you're watching a far-away supernova, the collapse and explode process will take a different amount of time in the supernova's own local rest frame than is seen by you. Taken alone, it's up to you whether you interpret this fact as time dilation of your clock relative to the supernova or a stretching of wavelengths due to cosmic expansion, or any combination of both.

However, what we have decided is to use a frame in which the homogeneity and isotropy of the universe is directly manifest, in which case we're not "looking" at a single supernova but all around us. For a general FRW spacetime, we cannot re-interpret cosmological redshift as due entirely to time dilation without spoiling those conditions.

The reason is that homogeneity and isotropy picks out a sequence of spatial hypersurfaces, i.e. a sequence of "nows", snapshots of the universe at instants of time. Then generally the distances between different galaxies are different at different times.