You do not need a special TV; a Roku device can make any "dumb" TV smart.. A Roku TV is already smart - i.e. it doesn't need a separate Roku device to make it smart.
You generally cannot get local stations. But if you have an HD antenna plugged into your TV, it's easy to switch back and forth between antenna stations and Roku stations. I do it all the time. If you don't want to do that, there are subscription services, like Sling TV, that include local channels, and can be streamed via Roku.
There is no DVR. I'm sorry I don't know if shows can be saved using a Roku and a recording device.
There is no monthly bill with Roku. However, if you add services like Netflix, Sling TV, etc. you will pay them a monthly fee.