Does the Church teach at what point a human being has an immortal soul?
Not exactly. It’s important to understand that the Latin word for soul is
anima. It’s where we get the words animal and animate, and inanimate objects. The soul is what makes something a living thing. According to Aquinas there are different types of souls and only human beings have immortal rational souls. He also speculated that the embryo starts out like a plant, then an animal, and then human in regards to soul, and there had been distinctions in Church law between the formed and unformed fetus in regards to the gravity of contraception and abortion. Quickening was considered an important time of transition for the soul (quickening being when the woman first feels her child move). More liberal pro choice Catholics will point to this in order to argue against the pro life position, though they dismiss how severely immoral Aquinas views contraception and even early abortion.
It’s really the discovery of the ovum and fertilization that changed part of the discussion. Before the understanding was that the man simply had seed which he planted into the woman. So as far as Aquinas was concerned, semen was more like a fertilized pre-embryo. Physicians in the early 20th century began arguing that a new human being distinctly comes into being at the time of fertilization which they called conception. The Catholic Church dropped its distinction between formed and unformed fetus, and quickening is no longer considered important as we now know the child moves long before the mother is able to feel the movements.
In the medical community though, the word conception got moved to implantation rather than fertilization, the word now only referring to the start of a pregnancy rather than the start of a new human life. Afterall, there are things like molar pregnancies (however rare) where only placenta tissue grows and there is no embryo at all.
When it comes down to the issue though, the big thing is that you’re not going to have a formed living human being without a human soul. If it’s alive, it has a soul.
Do tiny embryos go to heaven?
This is related to the question as to whether any embryo or fetus who dies before they can be baptized goes to Heaven. It used to be taught that unbaptized babies went to Limbo but this was later rejected as mere theological speculation. The Church then dug deeper into the tradition and the conclusion was basically “We don’t know but we put our faith in God’s goodness and good will.”