As far as I understand today we could use the field "Project Code" for that purpose.
Yes, you can do this !

Just take care than Project Code can also be used to calculate items reference.
You can also use the id : it is an automated number !
But when using such a nifty tool as Project'Or RIA it would be a shame to keep a separate list with used and unused project numbers - IMHO that should be handled within the poject management.
Why ?

Listing project will show you the project code, and the field is directly sortable, so you have the number list in Project'Or RIA.
I have no clue about how Project'Or RIA ticks inside. Coding in PHP, JS or CSS is not the problem. But is there any documentation or hint where to start when implementing new functions to Project'Or RIA?
No, not really, only phpdoc in the code is available.
So far I only found Dojo is used as JavaScript Toolkit, but what is the Framework for the PHP part?
You should not need to use Dojo.
The framework is a homemade one, quite easy to use : all you will need is in the /model directory.
For instance for your need you could :
- alter the database to create a new field (that could be an autoincremented integer)
- alter the Project.php code : add a property with same name as new field, at the place you want it to appear in the screen
- alter the save() method if you need to calculate the field
That's all.
Regards.