Since dark energy is causing everything to speed up infinitely, wouldn't everything near the speed of light? And since it's nearing the speed of light, wouldn't all kinds of strange things start happening such as time dilation?
The answer to your question is that if we assume the Universe is infinite then there will always be objects receding from us at the speed of light, simply because recession velocity depends linearly on distance. Indeed there are objects that we can see (i.e. they are within our observable Universe) that recede from us at the speed of light (this is a very misunderstood issue, so I'd advise anyone taking issue with this statement to make sure they've read this paper first).
If we assume dark energy takes the form of a cosmological constant, then the Universe asymptotically becomes like De Sitter space over time. This means that the sphere of objects that are receding from us at c (the Hubble sphere) settles down to a constant radius which is inversely proportional to the square root of the cosmological constant.
Edited to add: recession velocities though are purely a function of cosmological coordinates so don't confuse them with local velocities which are always constrained by c.
This is a tricky question, because dark energy does not always affect the distances between any two given objects in the universe. Consider this question: Galaxies are moving away from each other then how Milkyway and Andromeda galaxy coming towards each other?. The Milky Way and Andromeda are incredibly far apart, yet they are moving towards each other because of gravity. Likewise, the other galaxies in the Local Group are gravitationally bound to the Group, and won't be leaving it. The gist of this is that it's difficult to determine just where dark energy becomes the dominant player in the universe.
We can calculate the speed (recessional velocity) at which objects move away from each other by using Hubble's law, $v=H_od$, where $v$ is recessional velocity, $H_o$ is the Hubble constant, and $d$ is the distance between the objects. We can use the Hubble constant, along with the Friedmann equations, to derive the Hubble parameter, which can then be used to calculate the expansion of the universe over time. The Hubble parameter is a function of time, while the Hubble constant is constant throughout space-time; it is the current value of the Hubble parameter.
So while we can use the Hubble constant to calculate how fast objects are moving away from each other now, we can use the Hubble parameter to calculate at what time two objects will be moving away from each other at a given speed. I invite you to use the following pages to get you started on some calculations, but I will warn you to not use them for any objects inside the Local Group.
Here are the aforementioned pages:
I hope this helps.