To install Nix in a "non-standard" place, just:
$ mkdir -m 0755 ~/.nix
$ nix-user-chroot ~/.nix bash -c "curl -L https://nixos.org/nix/install | bash"

And to use it, issue:
$ nix-user-chroot ~/.nix bash -l

You're going to be in a user chroot where / is owned by your user, hence also /nix is owned by your user. Everything else is bind mounted from the real root.

The nix config is not in /etc/nix but in /nix/etc/nix, so that you can modify it. This is done with the NIX_CONF_DIR, which you can override at any time.