The higher the gain settings the more noise and interference. Sweep picking in particular needs lower gain settings, also try sweeping off the neck pickups. Different pickups have huge differences in tone so you will have to either tweak the amp settings or put an EQ pedal on front of it.
Forget the Spider presets, start from scratch and use your ears rather than concentrating on the dials too much. A pretty solid place to start tweaking from on your amp is with everything at about 6.5 on the dial - experiment from there.