https://nixos.org/wiki/How_to_install_nix_in_home_(on_anothe...
I've not personally tried it. From reading that document, it seems inconvenient but possible.
My understanding is that the hard-coded paths are needed to ensure compatibility of the binary caches, since store paths are often embedded in other artifacts (e.g. RPATH in binaries, shebang line in scripts). You could always run your own cache using another prefix if that's a problem.
http://lethalman.blogspot.ca/2016/04/cheap-docker-images-wit...
Other language managers don't hard-code and will allow stuff to run from a local directory.
Some rely on gcc, but again, it can be run from a local directory. Various dynamic libraries can be looked up by setting environment variables or passing parameters.
I think Nix is a great idea, but I'm not aware of an argument for hardcoding those root paths which truly holds water. It makes sense of NixOS, sure, but that's because it's an environment with top-to-bottom control.
The rest of us have to be good and share our toys.
As for why those paths have to be hard-coded: they don't it's a config variable you can change. When you change it you give up using other people's binaries so you have to build everything yourself, which is exactly like every other distribution system that relies on hard-coded /usr/whatever paths for binaries. The FHS standard is incompatible with Nix's guarantees and dropping it is where it gets all of its power, so it has to use something different, and it chose to make that path /nix/store/whatever
Some peers of mine recently experimented with building a Nix buildpack and concluded that it won't work. Nix has hardcoded paths for its core directories, and won't work in a security-constrained container.