The short answer to your question is yes, an FX lens will work fine on your D5500.
Another option to consider is the Nikon 200-500 F5.6. It has an excellent reputation for image quality and is a constant F5.6 aperture over the entire zoom range. It runs $1400 new, which is considerably less than an 80-400, and frequently is on sale for about $1050. On sale it runs a little more than the Sigma 150-600, but has some advantages over the Sigma, which has a good reputation nonetheless. I have used my 200-500 on a crop frame D3400 and it delivered excellent results.