No you can't sue just because you don't like the "noise". It's not America.
It has to be a nuisance, and not just something you don't like. It's a house doors away, and on the face of it its just a few social gatherings, and a chicken. The question would be if it was an actual nuisance, "Why has no-one else complained and why haven't the council done something about it under their powers?"
Is it just 'noisy'? A drum kit in the garden is quite a lot, at 2am!
Some councils are pretty bad, due to being a bit thick as well as costs cut. Been there myself with pubs which got seriously out of hand and to be honest, despite the police getting involved last year after a few very violent incidents that the council still haven't done much. They still haven't changed any terms to the licensing or made any effort to collect their own evidence but have installed CCTV in the street. But I do know that their 'out of hours' staff are available once every 3 weeks due to cutbacks...
Other councils, when faced with such anti-social behaviour and years of many complaints by the residents would actually start doing something because they do have the powers, as you say. Some councils just can't be bothered.
And yes, you can complain if music/noise is going on too late, too loud if it keeps going on and on. As I said, I personally wouldn't do anything with the courts but it is possible but leave it to the council as it's the same process. If the OP did take his neighbour to court, he'd need a lot of evidence where the noise is unacceptable in his home and it would be better if his neighbours got together because it's certainly strength in numbers.
I think too, unless you've been there that it's harder to understand that relentless loud music and noise into the small hours can get you down and have a real negative impact on your life. And while it's good to be a bit patient & understanding with noisy neighbours, why doesn't that respect seem to work both ways?
