The point is to also create a bit of privacy and I am not sure which tree is best for that.
Given your choice of Birch, Apple, and Elderberry, I'd venture to guess that you're somewhere in the eastern North America region. If you're looking for privacy, have you considered Spicebush or Serviceberry? These also grow in the same area and provide a decent amount of surface-level screening, possibly better than your current selection.
If you're looking for privacy, Birch actually isn't a great option. Its branches tend toward the top part of it, so unless you plan to prune it to keep it small, you'll eventually have an opening again and would need to underplant it with understory shrubs to provide privacy. (In any case, underplanting may be beneficial to your purpose. Have a look at the concept of "tree guilds.")
I am also wondering if an elderberry is a bad idea? I like elderflower and -berry juice, but I know that the berries can leave quite a mess.
That requires the berries to make it to the ground. ;) But seriously, birds love Elderberries, so while the berries can make a mess, they're more likely to get eaten before that happens. However, you can further mitigate that by planting far enough back from the sidewalk that pruning is feasible (so that it doesn't hang over the sidewalk) and plant herbaceous plants along the edge to help "catch" any stray berries. Coreopsis may be a good option in this case, since its dense growth creates a sort of "netting," its bright flowers help to attract pollinators, and it can tolerate full sun.
As for the question of how much space, that will probably depend in part on the variety of apple trees you have or plan to get. Elderberry can be just as happy as an understory shrub-like tree or as a standalone tree, so it can go closer to the apple than some other types of trees (even other apples). Just be sure that branches don't entangle so much that they can't get decent airflow, otherwise you risk fungal infections.