I need to handle an application for the 2006 version.
Starting with this version, BizTalk can route fault messages, and that's pretty good. Let's say I'm receiving a badly-formatted positional txt file and a receive pipeline can't let the message get into the orchestration. I can see that between the 'Suspended Services Instances' I have my suspended service gone in error with its code and its pretty explicit description.
I would love to get all this data into a file (txt, xml, whatever) and route it into a dedicate folder, without using custom pipeline components or other libraries, just BizTalk out of the box functionalities.
Is it all possible?

Yes, it is possible. If you set up Failed Message Routing on this receive port, you can create a filter subscription in another FILE send port for ErrorReport.ErrorType == "FailedMessage". This send port can save the error report and the original message out to a file.
Alternatively, you can have an orchestration subscribe to these failed messages if you want to do any further processing with them (e.g. logging of the error details).
See this article for some help with that, or view the official MSDN documentation here.