There is no correlation between live channels on cable and what is available outside cable. You don't channel surf.
There is probably 1,000 times more outside cable than on cable.
Here is Amazon's FireTV home page. Notice the boxes are providers that have menus and more content to show. At the top are Movies. Click that and a bunch of movies are shown from all kinds of providers. Click TV Shows, same thing, lots of shows from all kinds of providers. So, you can browse by provider or by movies or TV shows, etc.
Yes there is.
Recording shows is required so you can ff the commercials!