I called Chase and asked about this a few months ago. Here's what they told me regarding the Ink Bold:
They'll insure any portion of your trip that was paid for with the card (including flights booked with points, and taxes/fees paid with that card), up to the total value of the expenses for that trip that were paid for with that card. In other words, if you pay the $11.20 in taxes & fees for your award ticket with your Ink Bold, but also pay $800 for a hotel and $200 for a rental car with your Ink Bold, Chase will provide coverage of up to $1011.20 for any portion of that trip. That includes delayed flights, lost luggage, etc., even though you paid for the flight with points.
I didn't specifically ask about medical coverage, but it sounds like they consider this a covered trip, which would (I assume) mean that you'd have the standard medical coverage from any covered trip.
Of course, this was ~6 months ago, and things could have changed. I'd recommend calling to double check for yourself.