1. Your DVR capability won't be based on the streaming box you buy (AppleTV, Roku... etc.), but will rather be based off what streaming service you purchase (DirectTV Now, Hulu, Playstation Vue... etc.). Some have that capability right now and some don't. Read back through this thread and you'll see people discuss which DVR setup is best per streaming service.
2. When it comes to locals, I do not know of any streaming service yet that provides them, so as of right now, pretty much every cord cutter out there has to just get the locals through the air or use some type of on-demand watching after-the-fact through their channel apps. In my opinion, I've never gotten why people whine so much about antennas unless they are someplace that can't even get a signal even with the best outdoor antenna. Most people who whine about antennas use indoor ones, which is just straight up annoying and stupid unless you are forced to do so by living in an apartment. If you own a house, just pony up and get an outdoor one. I paid $50 for my antenna/mount combo and paid a guy $100 to install everything (he did a great job). One of the best investments I've ever made. For a one-time payment of $150, I now have local channels for life (did mine 8 years ago). Pretty tough to beat. With the setup of it in your house, now that you are a cord-cutter, what else do you use all your coax jacks around the house for anyway other than maybe one dedicated jack possibly for your cable internet? They now work perfect for hooking them all up to an outdoor antenna. Bingo.