sudo
why sudo can't work

how to fix it?
Unfortunately, you do not have access to root user without a rooted phone.
I have root
Then you are the first one to test this specific setup :)
I don't know if the proot'ed environment can handle the system root user. Another problem is probably the /etc/passwd which is currently staticly hardcoded. And for the sudo binary, there needs to be a wrapper similarly to https://search.nixos.org/options?channel=20.09&show=security.wrappers&from=0&size=50&sort=relevance&query=security I think. Feel free to hack around :)
#3
The closest thing I've tested to work was to run the failsafe shell, then su, then enter Nix environment with /data/data/com.termux.nix/files/usr/usr/bin/login.