It's alright. I do slightly prefer it to
Adrian. I like the historical connection with
Hadrian. It makes it seem much more solid and masculine than the more earthy modern
Adrian.
In the sibset you've listed, I actually think
Adrian would fit much better.
Nicknames for
Hadrian? Hmm... Haddy (with short a, not long) is the only reasonable sounding one that comes to mind.