It doesn't take much to get over-run with Raspberry plants. As you found out, they will root when the tip of a cane touches the ground. Planting them along a fence sometimes works best.
I wouldn't automatically prune them unless you have all the canes you want. Letting them go for a year or two wouldn't hurt anything if you have an area big enough and you have enough room between plants so you can pick them. Pruning only the tips that touch the ground might be the best plan until the main roots are situated and mature. You can then plant the tips you pruned off in a place where you want them. Raspberrys also like fertilizer in the spring. A half pound of 10-10-10 per plant, about 12" away from the established base will give them a good start to the growing season.
Picking the first year won't hurt the plants either, but as you know, mature plants produce more. As a rule, I don't pick the first season, but I do have a few 'taste tests'.
I doubt the winter will kill the younger ones off. Raspberry plants are very hardy. What you can do is put the younger ones in a pot and keep them in a relatively cold place, covered with leaves or other light mulch until spring. Then plant them where you have room. Leave them be if they already rooted themselves in the area, unless you don't want them there. You can transplant them in spring. I have no doubt that in a few years you will wonder why you were so concerned about getting raspberrys started. Because in 3 years you should have all the raspberrys you can handle.