invoke zsh, having it run a command, and then enter interactive mode instead of exiting

Not that I would advise doing this.

(sirius)~: zsh -c 'print hello; zsh -i'
hello
(sirius)~: echo $SHLVL
2

There are other tricks you can play with screen and using the $STY variable.

If you want something run from zsh with individual screens, you can check the $STY variable within your .zshrc or .zlogin. It is in the format <PID>.<TTY>.<HOSTNAME>.

if [[ -n $STY ]] then
  if [[ -f ~/.zsh-$STY[(ws:.:)2] ]] then
    . ~/.zsh-$STY[(ws:.:)2]
  fi
fi

If in screen, and if ~/.zsh-<TTY> (from the $STY variable) exists, source that, then continue on your merry way. You can also set an environment variable before calling the interactive shell.

> FOO=bar zsh -i
> env | grep FOO
FOO=bar

> RUNTHISCOMMAND=/path/to/script zsh -i
.zshrc:
if [[ -n $RUNTHISCOMMAND ]] then
   $RUNTHISCOMMAND
fi

Add those checks into your .zshrc/.zlogin.

I found a solution that works without an extra shell here. Add:

if [[ $1 == eval ]]
then
    "$@"
set --
fi

to .zshrc, then call zsh with

zsh -is eval 'your shell command here'

Really great for starting up lots of shells at once.

I have eval "$RUN" at the end of my .zshrc. I can now run commands without the extra shell, with:

RUN='my_prog opt1 opt2' zsh