Is it possible that in these times, the amount of energy would permit atoms larger than we have observed in nature or been able to create in the lab?
Quite possibly, but we do not need to go back in time. In fact, the early moments after the Big Bang are the least likely time to find very heavy elements, as the primary elements produced initially are believed to have been almost exclusively hydrogen and helium.
Since those early moments, the universe has certainly expanded, but there are still places where energy and mass are highly concentrated, such as in supernovas and black holes. Whether or not very heavy elements are transiently created in these places is an open question.
The problem is that elements heavier than uranium tend to be unstable, so they break apart (nuclear fission) very quickly into lighter elements. In order to make very heavy elements, you need to combine two less heavy elements (nuclear fusion), but it's hard to get two heavy atoms together to fuse if they keep falling apart themselves. A good discussion of the possibility of very heavy element creation in stars is
Goriely, S. and Martinez Pinedo, G. (2015) The Production of transuranium elements by the r-process nucleosynthesis. Nuclear Physics A 944:158. DOI:10.1016/j.nuclphysa.2015.07.020
If we then travelled to a far future, hundreds of millions of years from now, when the energy of the universe is much more spread out, and atoms have had more time to decay and lose electrons, wouldn't there be, at some point, a time where we no longer observe certain heavier elements in nature?
The answer to this question relates to the first answer. Since heavy atoms are constantly being synthesized within stars, the elements that are decaying get replaced. But as the number of stars decreases over time, the rate of synthesis will necessarily decrease as well, and elements that are susceptible to fission will slowly decrease in abundance. The time scale for this, however, is much longer than even hundreds of millions of years. Our Sun, for example, is expected to last about another 4.5 billion years, and the final dying out of all of the stars in the universe is expected to take somewhere between 1 and 100 trilliion years. I encourage you to read up on ideas about the "heat death" of the universe if this is a topic you are interested in.
At either of these points in time wouldn't you expect to find more ionic elements in nature?
Since it takes energy to separate opposite charges from one another, ionization happens with high energy, so places like stars where there is a lot of energy will have a lot of ionic matter. Likewise, when high energy emissions from those places interact with other matter (like cosmic rays hitting the Earth's atmosphere), ionization will take place. So you're right that when the universe's energy is more concentrated, there were probably more ions.
In the distant future, the opposite is true. Since the universe decays to a low energy state, and ionization takes energy, we expect that oppositely charged matter will combine to make neutral atoms and molecules as the universe gets older. Some have theorized, however, that the expansion of the universe will mean that matter is so spread out that oppositely charged ions get too far away from each other to interact, so it is conceivable that the final state of the universe is separated ions, but that is just speculation.
And as noted in the comments, you may get a much better answer over at physics.SE.