Learn English – What tense is this sentence in? “Having had”

You say in the comments that this sentence describes an "ongoing development project". I assume then that the context is an interim report of some sort, in which you are reporting the project's current status.

That being the case, you should probably cast everything in the present perfect, whose fundamental sense is a current state arising out of past eventualities. For the adjectival clauses you may employ participle clauses rather than full relatives:

This document describes the programming conventions used in the library development, the library directory structure, and the supplementary tools developed to support and manage the library development. The main part of the library has been written in C++.

But nobody would quarrel with you if you wrote that the main part was written in C++.

It's pretty wordy, thanks mostly to a bunch of clunky nominalizations, and it's needlessly passive. I'd be inclined to write:

This document describes the programming conventions, directory structure and supplementary tools we've used so far in developing the library. We've written the main part of the library in C++.

—shorter by nearly a quarter, and a lot friendlier and easier to read.

"Have been having" isn't a single phrase, it's part of "have been having trouble", which contains two parts and two different meanings of "have".

In the first half, "have been" expresses action that started in the past and continues to the present.

In the second half, "Having" means experiencing, i.e., experiencing trouble.

I am having trouble sharing my neighbors wireless connection.

means I am currently experiencing trouble sharing my neighbors wireless connection.

I have been having trouble sharing my neighbors wireless connection.

means that trouble sharing my neighbors wireless connection is an ongoing situation that I started to experience in the past and am still experiencing now.