Launch a read-only web server displaying some sort of shell output

You need to do it in a part:
The script below will launch a shell output recording session.

echo /tmp/$STY
touch /tmp/$STY
chmod 0600 /tmp/$STY
script -a -f /tmp/$STY
rm /tmp/$STY

Then you need to run the following:

shellinaboxd -t -s /:$USER:$USER:/home/$USER:"/usr/bin/tail -f /tmp/$STY" &

You can put it in a script, as $STY is exported. Please don't single-quote ('$STY') that variable.