To your questions.
Yes you can spin fish.
Yes, it is an excellent technique.
Yes, the area, as many trout waters are, is popular with fly anglers, due in part to the nice sized fish.
Yes, you should do well with spinning gear, provided you're flexible and work a variety of lures/techniques.
Yes, shore fishing works, with warmer waters, evenings/mornings are likely to be better options.
Love my float tube, as it is slow paced and give good access to the water, but there are other excellent choices. Personal preference on this one.
Others will have to help with the lures, but if it was me, small tubes, marabou jigs, inline spiners, spoons, and small minnow (up to size 5 or 7 rapala) would all be in my arsenal. Consider also carrying a few "bubbles" and flies another good technique with a spinning outfit.