No, you can set max_execution_time to any value you want. I have had PHP scripts running well over an hour in time.
I don't understand this part completely. You want a section inside your web page that loads content generated by your object code, but you want the rest of the page to render first in case it takes a while? If that is the case, you could do something like that with AJAX and I don't know how an iframe would work.
As for executing PHP code after outputting content, that is a feature that PHP is lacking. It is unfortunate because a lot of people would like to have their scripts continue executing without making the browser wait for PHP to close the connection. With a PHP-FPM setup, you can use fastcgi_finish_request() to do this. But with a standard PHP setup there is nothing that I am aware of that works and I have seen a lot of proposed solutions over the years. I never tried this but you may already have:
If a program is started with this function, in order for it to continue running in the background, the output of the program must be redirected to a file or another output stream. Failing to do so will cause PHP to hang until the execution of the program ends.
See the examples that follow in the comments section.