I'm probably thinking this is easier than it actually is, but it reads like you want to take all the data in your current request-details page and create a new record in your projects table using that data.
If the button that will create the new record is taking the user to another page, I'd recommend coding the current request-id value into the button via a hidden form input tag:
<input type="hidden" name="request_id" value="<?php print $request_id ?>">
The next page will see that value and you can use it in a select query to get those details values again based on that request_id
Then, write sql to insert those values into your projects table. You just have to pick out which values from your details record you need to use in your insert sql to match up to the fields in your projects table.
Hope this helps.