They're very close to each other that the features that one may have over another is far more important.
Here's a comparison of a few models of Nikon and Sony - http://www.dpreview.com/reviews/image-comparison?attr18=daylight&attr13_0=nikon_d3300&attr13_1=nikon_d5300&attr13_2=sony_a6000&attr13_3=sony_a3000&attr15_0=raw&attr15_1=raw&attr15_2=raw&attr15_3=raw&attr16_0=100&attr16_1=100&attr16_2=100&attr16_3=100&normalization=full&widget=1&x=-0.23389412034910426&y=-0.6256653633746471
You just don't see all that much of a difference. However, put a fantastic high-end lens on just one of them, and then that will be the with the better image quality. With all things being equal, your lens is going to be deciding factor when it comes to sharpness. The features in the body such as high ISO performance, shooting frame rate (fps), AF speed, lag time between pressing the shutter and the camera taking the shot, built-in IS, and other things will all have an impact on image quality. The Nikon D5xxx series has a larger dynamic range than the D3xxx series because it shoots 14-bit RAW files compared to the D3xxx 12-bit files.
Any DSLR will focus faster and better than a mirrorless. You buy the Sony A series for it's advantages over the larger DSLRs and vise versa.
Before you buy the D5500, take a look at this:
https://www.youtube.com/watch?v=a8Je6h_c_tQ
Here's a side-by-side comparison of the models you've listed, and I've added in the Pentax K5IIs and the K3 to show you that there are better bodies than Nikon. In APS-C game, Pentax is considered to be the best around -- http://www.dpreview.com/products/compare/side-by-side?products=nikon_d3300&products=nikon_d5500&products=sony_a6000&products=sony_a3500&products=pentax_k5iis&products=pentax_k3&sortDir=ascending