Where should this be set? Inside shell.Run or maybe just main?

In general the only time a shell should set the SHELL env var is if it isn't already set.

I empirically confirm @krader1961 is correct. I just checked what SHELL is if run inside a bash run from zsh (which is my default shell):

❯ echo $SHELL
/bin/zsh
❯ bash
...
~▶ echo $SHELL
/bin/zsh
~▶ 

So this is consistent at least with bash. I also tested nushell, and it also doesn't change SHELL. So the question is if SHELL is set if elvish is set as the default shell?

After temporarily changing my login shell with chsh to /usr/local/bin/elvish:

|> echo $E:SHELL
/usr/local/bin/elvish

So it seems the value is being set correctly, to the user's login shell. Note that the SHELL environment variable is normally set by the login process, so the shell does not have to do anything. See for example http://man.openbsd.org/login.1#ENVIRONMENT (the macOS login man page says a similar thing).

Also, the SHELL env var is not meant to reflect the current shell. It is meant to reflect the shell that programs like make and vim should use when running "shell" commands if they haven't been explicitly configured to use a particular shell. That is normally the user's "login" shell which is historically expected to be at least minimally POSIX 1003.2 compliant -- which Elvish is not. This is one reason the recommended way to run Elvish is not to change your system "login" shell to Elvish.

I actually think Elvish should never explicitly set the SHELL env var. Not even if it is not present when Elvish starts. The path to the running Elvish binary should instead be exposed via a variable in the platform: namespace. The user can then choose to assign that to the SHELL env var (presumably in their ~/.config/elvish/rc.elv script) if they have decided that is appropriate for their situation. The var might be named platform:elvish-bin-path.

In bash, sh and zsh, the argument zero $0 points to the shell executable that was run. Even though it can be the name or absolute or relative path (bash or /usr/bin/bash), but still it's useful (for example to write scripts that work in several shells, or behave differently according to shell).

But in Elvish, $args is empty by default and lacks the "argument zero".
So I suggest adding a new variable like $executable that points to Elvish executable (full path for consistency?), like Python's sys.executable.

In bash, sh and zsh, the argument zero $0 points to the shell executable that was run.

That's only true for interactive shells, those run with the -c option to execute a set of statements on the command line, and those that read statements from stdin. For shell scripts it is the path to the script. The latter case is handled by the src command in Elvish.

So I suggest adding a new variable like $executable that points to Elvish executable (full path for consistency?), like Python's sys.executable.

I don't have a strong opinion (at this time) for whether this should be exposed as builtin:executable, platform:elvish-path, or something else. But, as I said earlier, Elvish should never set E:SHELL to its path.