Are there predictions of the distribution of matter at the end of time Are there any theoretical predictions or "educated speculation" by cosmologists on how matter will be distributed at very, very distant times in the future?

For example, will matter be:


agglutinated into a big cold ball?
diluted across the universe in tiny bits?
large clusters of matter here and there?
something else?
The basic answer to your question is "yes to all of the above, in different eras".

If we use the concordance $\Lambda$CDM model, with the cosmological constant remaining constant, then the universe will keep on expanding at an accelerating rate forever. The first big effect of this is that gravitationally bound clusters of matter such as galaxy clusters will become separated by ever larger distances (but they themselves are not changed noticeably by this). The current cosmic web of galaxies and voids disintegrates into little separated droplets over the next few hundred billion years.

Inside these "island universes" local gravitational interactions tend to make galaxies merge with each other, eventually ending up as a single big elliptic galaxy.

Over long time (around $10^{20}$ years) close-encounters between stars lead to some stars being flung out of the galaxy into the outer darkness, and the others bunching up closer to the central black hole. Eventually this reduces the galaxies to big black holes and scattered stars. The scattered stars cool down to low temperatures.

If proton decay happens, then somewhere beyond $10^{33}$ years or so the scattered stars quietly and slowly evaporate into elementary particles. The black holes are also expected to evaporate, and after $\sim 10^{99}$ years or so the only thing left will be separate stable elementary particles - electrons, positrons, neutrinos, photons, maybe some stable dark matter. Due to the expansion each particle will be alone in its visible universe. 

A good overview is Adams and Laughlin's "The Five Ages of the Universe" (or their paper, for more detail).