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Yes. Look for a television that has a CableCARD slot. Here is an example:

Pioneer Elite PRO-930HD

Note that very few televisions are available with CableCARD slots, because your neighbors are not willing to pay the extra money associated with providing such a feature. CableCARD slots in televisions were much more common a few years ago, but practically no one bought them, so manufacturers listened to their customers and stopped making them. As a result, you have many fewer choices available to you.

Pretty-much. You can connect a VCR's "Video In" ports to many digital devices and do manual recordings, and connect cables from a VCR's "Video Out" ports to "Video In" ports in many digital devices, but the tuners inside VCRs are antiquated technology that won't work on digital signals. Even with manual recordings and playback from VCRs, you'll be exceedingly unhappy with the result, because VCRs are incapable for providing signal that looks as good as a digital signal can look on a digital television.