2. Am I right to assume that I need a convertor box with the simple antenna to pick up the digital stations as well?
Only if your TV does not have an integrated ATSC tuner. The coaxial input of a TV with only NTSC support vs a TV with NTSC & ATSC looks exactly the same, it is a single input.
Since your TV is only 5 years old I would double check if there is not already an ATSC tuner in it. Just doing a channel scan when you get your antenna will give you an idea, it should display digital channels if it sees some.
EDIT: there may be other device you already own which also have an integrated ATSC tuner...