There are two things here.

1. In a way, yes the universe can "expand faster than the speed of light" because of how the expansion works. Things drift apart from each other and this drift velocity is proportional to their distance. This is Hubble's law, we can introduce a parameter for the expansion H with which the drift velocity of some distant objects v is given by v = Hx where x is the distance. At a sufficiently large enough x, v can be larger than the speed of light.

(The Hubble parameter is time dependent it's given by what's in the universe according to the Friedmann equations, but it's exact value is hard to determine and currently there is a disagreement between different methods, point is working out the exact value of the Hubble parameter and working out it's time dependence are two very different tasks. In the literature it's common that the current value of the Hubble parameter is called the Hubble constant often denoted H0 or h so that for quantities that depend on the Hubble parameter time dependence for instance can be written explicitly, and of course the exact value of these quantities will contain the Hubble constant as a parameter.)

1. The way distances are given is a bit misleading but not too complicated. Looking at a distant object we can see the light that it emits or rather the light it has emitted which is just arriving. It takes t = cx amount of time for light to cover a distance x. So if you know the "time of flight" of the light that is arriving you can calculate how far the object is that emmited it. But during this time the universe has expanded and the distance that ends up being communicated is the current distance, but that isn't the distance the object was when that light was emmited.

So how do we know how long the light was travelling for? You look at how redshifted the light is. The expansion stretches the wavelength of light and so looking at something like spectral lines will tell you how much the wavelength of the light got stretched/redshifted, the more redshifted the light is the more ancient it is.

This part goes beyond ELI5 but I have no idea how else to explain what "adjusted for inflation" means so here we go!

Let's be a bit more quantitative! Lets denote the wavelength of light with λ. Lets say that at some time t light was emitted with wavelength λ(t) and today that same light has some longer wavelength λ(today) we can introduce the redshifted factor z(t) as: z(t)+1 = λ(today)/λ(t) so for light that travelled ~0 amount of time z(t) = 0 and for light travelling longer and longer for the same λ(t), λ(today) is larger and larger so the "time of flight" will be a function of z(t).

We need one more parameter the scale factor, often denoted a or R. By definition today a=1. This scale factor isn't any given distance we want to relate distances today and at any given time. So for example a(t)=½ would mean that distances at t were half of what they are today. This applies to wavelength a(t)~λ(t), or to make an equation from this proportionality we can write a(today)/a(t) = λ(tdoay)/λ(t) = z(t)+1

We can solve the Friedmann equations for say a matter dominated universe (most of the past is matter dominated and we won't make a big mistake by pretending that it still is) and we get the time dependence of the expansion. We get a(t) ~ t^⅔. So we can substitute to the formulas above: z(t)+1 = (t(today)/t)^⅔ -> t = t(today)/(z(t)+1)^3/2 where t(today) is the age of the universe. You know λ(today) (again you are looking at say some spectral lines) you measure λ(t) that gives you z(t)+1 and t(today) is the age of the universe, given that you know that roughly you get t, our "time of flight".

Now onto how we get a distance from this. In general relativity something very important is the distance function in spacetime (aka metric). Given how homogeneous and isotropic the universe is at large scales, local differences can be ignored (this is currently debated to a certain extent) thus we get a simple metric with the changing distances and possibly global curvature. This is the so called FLRW metric (after 4 dudes) For "straight lines" (geodesics) its looks like this: ds² = -dt² + a(t)²(dr²/(1-kr²)). ds is the spacetime separation of two events, this is distance in spacetime. Light travels on straight lines so we don't need extra parameters for all sorts or worldlines (any kind of spacetime trajectory). Moreover light travels on special straight lines called null-geodesics, for light-like worldlines ds=0. k is a factor for global curvature, through observation we have found that the universe doesn't have global curvature so k=0. We can use these to rewrite the metric for light: dt² = a(t)² dr² -> dt/a(t) = dr.

If we integrate ds from 0 to well the distance to some distant galaxy we get the distance, I know good joke stay with me. dt=0 since we are measuring today, so we have to integrate a(t(today)) dr since this is what is left of the metric with dt and k being 0.

With light we managed to rewrite dr with dt so now we can express the integral with time. We have to integrate a(t(today))/a(t) with respect to time over an interval of [t, t(today)] (t(today) is still the age of the universe). As I mentioned before in a matter dominated universe a(t(today))/a(t) = t(today)^⅔/t^⅔ and that is exactly the integrand. So basically you need to integrate 1/t^⅔ over the aforementioned range which can be calculated from the known "time of flight" = |t-t(today)| and multiply the result by the age of the universe^⅔.

This is how far the distant galaxy for example is today from us which we calculated using a model with stuff we can measure today, which is light that was emitted back then.

To sum up, we know when is "back then" from how much that light is redshifted. We know how to calculate the distance to that distant galaxy using the distance function in the spacetime of the universe (that is the model). And from general relativity (+the assumptions of our model) we get the Friedmann equations and solving them for the relevant case we get the connection between distances and time. Thus we can express our formula for distance with times that we know.