Christians who accept evolution don't necessarily agree with each other. For better or worse, this approach is sufficiently new that there's no agreed doctrine of theistic evolution.
Yes, there is. See the quotes in my signature.
Evolution may only have been well-articulated in 1859, but Christianity has grappled with the idea of conflicts between extrabiblical evidence and scripture from the very beginning. Augustine of Hippo in the 400s articulated the basic idea of theistic evolution: scripture cannot contradict God's Creation. If there is an apparent conflict, then the interpretation of scripture is wrong.
John Calvin in the early 1500s in is
Commentary on Genesis restated this. The Catholic Church faced the problem of what to do when the church incorporates a scientific theory as part of its theology when it confronted heliocentrism of Copernicus and Galileo. Although it seems that the Catholics came down on the side of the primacy of scripture, in actuality it was the reverse. Never again would the Catholic Church deny extrabiblical evidence that contradicted an interpretation of scripture.
Those who are closest to a literal reading of the Bible would say that there is a literal Adam and Eve, who were the first with human souls. As you move from that, you get into complex areas where I don't think there is agreement. Examples:
None of the examples are part of theistic evolution nor are unique to it. Instead, they are questions that get asked whether God created by creationism or by evolution.
The fallacy in your thinking is the premise that souls are material and, therefore, evolved. However, standard Christian doctrine is that souls are
immaterial and infused by God. Souls are not part of evolution.
It would help if people who think about evolution would actually
read Darwin. Darwin addressed the issue of souls and put souls outside evolution:
"He who believes in the advancement of man from some low organised form, will naturally ask how does this bear on the belief in the immortality of the soul.... Few persons feel any anxiety from the impossibility of determining at what precise period in the development of the individual, from the first trace of a minute germinal vesicle, man becomes an immortal being; and there is no greater cause for anxiety because the period cannot possibly be determined in the gradually ascending organic scale."
Literature.org - The Online Literature Library The Descent of Man
If so, then there could have been a continuous development, where the line between animal and human is somewhat arbitrary.
Humans
are animals. In anthropology today, the term "human" is applied to any species in the genus "Homo".
Suppose Adam and Eve represent, not the first time there is some new part or capability, but the first time God dealt with humans as responsible creatures.
This supposes that Adam and Eve were real. They were not. They are allegorical figures.
God doesn't have to put a soul into just 2 individuals. God can put a soul into each individual of an entire population at the same time. So, sometime in the evolution of hominids -- we don't know when -- God puts a soul into each and every individual alive at that point. No big deal.
I suspect that animals have, to varying degrees, some of the same capabilities as humans, although as noted above I think currently there is a significant difference between humans and even the closest animals.
Can you specify that "significant difference"? Biologists have been trying for quite a while, but every time they come up with an idea -- such as tool making or language -- that is supposedly unique to humans, it turns out that other species have it.
I submit that the "significant difference" you see is not biological, but technological. You are making the same mistake lots of people have made: thinking that a difference in technology means a biological difference.
Instead, consider the idea that humans have 2
small differences from all other species: the ability to make tools to make tools and fine control of speech so that there are nuances to their vocalizations. From these 2 differences arise all human technology. The fine vocalizations offer better communication of ideas since a wider range of vocalizations permits a wider range of words to describe more ideas.
This saves us from the problem of saying that there is some magic point in evolution where suddenly a soul appears as a physical or metaphysical part or capability.
We don't have to say that unless you assume that soul is a material entity which has to evolve. In the Adam and Eve story God infuses a soul directly. Why couldn't He have done so at some point in evolution?