This is a nonsensical question; it isn't possible to have reproduction without evolution.
It's possible to have reproduction without evolution.
We have monoclonal immunocompromised mice with fixed genomes. We created them.
One evolutionarily change I predict is that certain forms of birth control will become less effective over the next generations. Hormonal birth control isn't 100% effective and there's probably a genetic component to that. So the women for whom birth control doesn't work as well will tend to have more offspring and pass those traits on.
Of course it will. For example, social media has completely changed the way people date and pick partners. This will inevitably change the next generations.
A good example is sickle cell trait, which provides a higher resistance to malaria, but a risk of a variety of other complications. In areas where malaria is a severe threat, the survival advantage of the higher resistance outweighs the other drawbacks. Medical intervention such as administering a malaria vaccine might reduce the threat of malaria such that there would be selection pressure against the sickle cell trait instead of for it.