would a 550 watt PSU work on my build?
This is my build: http://pcpartpicker.com/p/xxYY3C
Yes. It's recommended to have atleast 550w PSU when using r9 380, which you are planning to get.
Yes, its plenty, you could probably use a psu that is about 150 watts lower.
Yes it will and you can even get a 500WPSU. If you want a 80+ Gold Certified PSU, then I would recommend http://pcpartpicker.com/part/evga-power-supply-220gs0550v1 . If you want to save some money, get a bronze certified PSU. For a bronze certified PSU, I would recommend http://pcpartpicker.com/part/corsair-power-supply-cx500m . The bronze one is $52.98 while the gold one is $70.98. Hope this helps.