I'd be glad, if there could be a new feature in MO.

It happened to me several times, that I commented to a question or an answer, and somebody commented upon my comment. then I just rediscovered my comment browsing through the question, but my response upon the comment response was probably to read by the person I adressed it to.

Is it possible to include a feature, which informs you about all consecutive comments given to a comment of yours?

Not just because of you keeping track of comment responses, but also to get a reaction on comments to another comment.

It also'd be nice to have a overview about one's comment given...

Of course, there is still the issue with editing comments...

For keeping an overview (at least for recent comments), you could use your userpage (choose recent, instead of stats). For the notification (as some sort of workaround), you could use the RSS feed of the question's page (lower right corner).