Basically, you've got the right idea.
You want to use something more like winter rye or hard red wheat or field peas, or a mix of all three. If you wanted to harvest a crop edible by animals, you could go with clover, field peas, oats, turnips, Russian mustard greens. I don't think I'd use alfalfa, because to get a good alfalfa hay crop takes a lot of work (seeding twice, takes a long time to grow, young-ish plants don't tolerate frost well, has to be cut just so), whereas the other stuff you can grow any old how and it'll do fine. I would put at least one legume (field peas, clover, vetch) in there--they fix nitrogen and you won't have to add nearly as much fertilizer in the spring.
Cover crops are meant to be discarded when you're ready to plant in the spring--either tilled under or cut for fodder and then the remaining bits tilled under. So no, it shouldn't delay spring planting, unless you don't find time to do the extra work of cutting, drying and tilling. The only reason spring planting would be delayed is if you were intent on getting a fodder crop from it, and you wanted to wait until later in the spring to get a higher-quality hay. Most hay crops (clover, alfalfa, etc.) are best cut when the tops are just budding and not yet flowering--which won't happen until late spring/early summer. If you've got started plants in pots ready to go in the ex-hayfield, that's fine, but if you wanted to seed direct, then the hay crop will delay anything else in that field so long that you may only be able to plant short-season stuff there.