The theory doesn't predict how evolution will affect populations because - as you pointed out earlier - it is the environment that selects which random variations will be most successful. We can't predict either the environmental changes on evolutionary scales of millions or billions of years, or the random variations in individuals or populations.
Given the caveat on predictions (above), it's impossible to say in advance, but with hindsight the evidence is clear - and it is entirely consistent with the ToE.
Well that's just one implication of the ToE. Darwin formulated the theory by looking at a wide variety of extant life in a wide variety of geographical locations and environments and considering why they followed the patterns they did. Since then, every discovery has confirmed his idea, especially the millions of fossils discovered, and the genetic relationships between extant life.
But at the time, there was no molecular biology and relatively few fossils, but even so, it was possible to see that all vertebrates had the same basic structure and body plan, from skeleton to organs, all made in the same way, and it was possible to see that, for example, all mammals shared specific differences in the basic body plan, and the more similar the species appeared, the more similar their internals were. IOW they gave every appearance of being related.
What's more, it was possible to see how small changes in an ancestral tree from a common ancestor could produce those similarities in every mammal. The same process could be applied to comparisons between vertebrate classes - and the clues were there - fish that could walk along the seabed using modified fins; walking and hopping fish that could spend time out of water, gulping air to breathe; amphibians that spent most of their time out of water, reptiles that were wholly land-dwelling, and so on. All seemingly displayed progressive anatomical steps that related them by common ancestry.
Then, when they looked at the embryological development of all land-dwelling vertebrates, they found clear indications of shared ancestry with fish - the early stages of embryo development were almost identical, they all developed tail-like features, all tetrapods developed slits ('pharyngeal arches') below the head, and limb buds. Then they began to diverge; the slits became gills in fish embryos, and face, jaw, and ear structures in mammals; the limb buds became fins in fish and limbs in mammals, etc. This strongly suggested a
common ancestral origin.
As time went on, more and more fossils were found, also consistent with the ToE, and eventually, molecular biology (modern genetics) showed that the same patterns of relationships were present in the genes. It showed that the closest genetic relation to humans was the chimpanzee, and we found fossil humans and hominins that led back through a complex lineage to a primitive ancestor in Africa, all consistent with the ToE and the genetic, embryological, and other indicators of
common ancestry.
So it helps to see the evidence for the bigger evolutionary picture to get an idea of the overwhelming amount of evidence for it, not just 'fish to humans' - and I've only mentioned three lines of evidence, there are various others. For a full and readable description, I would recommend Neil Shubin's book, '
Your Inner Fish' (seriously).
Well, PE was proposed to explain the 'sudden' appearance of species in the fossil record, and is generally thought to have been rather exaggerated by Gould. But it has been tested in the fossil record by finding fossils that did, as it predicts, change very little over evolutionary timescales; also, more species were found that appeared relatively suddenly in the record, but that had clear ancestral precedents at various temporal removes.
The main explanation for periods of relative evolutionary stasis is the lack of selective pressures, i.e. a consistent environment. Species change rapidly or go extinct when the environment changes relatively quickly. There are living creatures that have changed relatively little for millions of years and we find that they are either extremely adaptable (e.g. ants), or their environment has remained relatively stable for millions of years (e.g. oceanic fish, particularly deep ocean). We also find that relatively rapid environmental changes lead to rapid observable evolutionary changes.
The ToE is tested almost every time a new discovery is made in any of the lines of evidence that support it - it could be falsified by almost every new discovery if it didn't conform to the expected pattern; every new fossil discovered, every new species whose genome is sequenced, every new embryological study, and so on. The classic example, given by J.B.S. Haldane, was "
fossil rabbits in the Precambrian". As it happens, we've had ~150 years, with millions of new discoveries, including in whole new fields of biology that didn't exist when the ToE was formulated, and they all support it, without exception ¯\_(ツ)_/¯
Caveat - to whom it may concern, I wrote this in a hurry, so if I made a blunder, please point it out
