Presumably, all these directories are part of your PATH environment variable. If you remove them before running dune, it should get rid of the warning messages, e.g., by typing PATH=/usr/bin in your console. (You might need some more directories than just /usr/bin in your PATH. Type echo $PATH to see which are the currently existing ones.)
I am not sure what your point is; you have just showed that /etc/environment indeed exists on your system, but neither that it is free of /usr/games nor that it is ignored by Snap. That being said, the content of that file might be irrelevant, because Snap itself presumably comes with its own /etc/environment file (under /snap/coreXX/current/etc/environment) which would override the system one, I guess.
$ for dir in $(cut -f 2 -d "=" </snap/core20/current/etc/environment | tr ":" " "); do ls -ld $dir; done
ls: cannot access '"/usr/local/sbin': No such file or directory
drwxr-xr-x 2 root root 4096 Dec 28 04:52 /usr/local/bin
drwxr-xr-x 2 root root 12288 Dec 31 07:31 /usr/sbin
drwxr-xr-x 2 root root 28672 Jan 4 16:37 /usr/bin
lrwxrwxrwx 1 root root 8 Dec 28 04:52 /sbin -> usr/sbin
lrwxrwxrwx 1 root root 7 Dec 28 04:52 /bin -> usr/bin
drwxr-xr-x 2 root root 4096 Dec 9 21:08 /usr/games
drwxr-xr-x 2 root root 4096 Dec 28 04:52 /usr/local/games
ls: cannot access '/snap/bin"': No such file or directory
But:
why in the world should coqc use PATH to look something up? (What “something”?)
The PATH contains of nine different parts. Why are only three warnings logged in each call to coqc? Why are some logged which are in path an can be accessed (e.g. /usr/games) and other with are in PATH and do not exists are not logged (e.g. /usr/local/sbin)?
Coq does look up in the PATH for some helper tools, e.g., csdp, which is used by the nia and nra tactics. (The odd part is that the lookup is always performed at startup rather than being delayed until the tactics are actually used.)