Organizing technical documents and reports
It might help to add a few details to enable specific constructive responses. What is the purpose for organizing the documents: personal use, internal use within your team/department, public but less formal use such as your personal webpage, public and more formal use such as website of your research group, or inclusion in CV? 

Each of these uses is aligned with different (though possibly overlapping) audiences, and might imply different approaches for what and how to present, what to highlight or omit, etc. etc. 

So, first I recommend to organize your organizing needs/priorities, then address details.

A few general ideas, with regard to organizing in "professional/formal manner":

Personal use: There are tools such as Mendeley that can be helpful for figuring out and experimenting with ways to categorize the documentation. Mendeley can be leveraged for open/public access to materials as well as private use (e.g. uploading citations but not actual attachments).

Research team website: collaborate with your team to determine what might be worth sharing publicly (beyond what has already been made available). For instance, you might collectively decide that adding some previously un-shared white papers/tech reports, or even datasets might "fill out" the public profile of the group's research and allow the team to cite/reference these resources.

Personal website: Regardless of whether you make the material available on a collaborative formal webpage of your department/group, I see no problem with also featuring a few technical reports/white papers that you have authored or co-authored on your personal department page. Technically, if you work for a public university and unless your research is classified, your research output is in the public domain and you should be able to enable public access to it without any negative consequences. (There may be other specific caveats around this in each case, but you need to explore those and use your best judgment).

CV: In my experience, citing technical reports or white papers in your CV is totally acceptable and could serve to boost your research track record. Since tech reports are not typically externally peer reviewed, a safe approach is to place them in a separate dedicated section, e.g. "Technical Reports."

Finally, there are numerous sites out there such as Mendeley, Academia.edu, ResearchGate, etc. The sites differ in functionality, target audiences (e.g. more hard vs. more social sciences) and visibility/uptake within a given discipline. Here is a fairly complete listing as a starting point:
http://connectedresearchers.com/online-tools-for-researchers/. Good luck!