Yes.  You have to claim your luggage when you land in Belize City and
do the customs/immigration thing.  Then you can go to the puddle
jumper counter and check in for the flight that you can book ahead
online if you like.  Supposedly it's not necessary to book ahead since
they make the flight every hour or so and when they have more people,
they simply upgrade to a larger aircraft.  They are very flexible.  On
my return from Ambergris, the flight actually took off about twenty
minutes before the scheduled time.

While waiting in Belize City, have a rum punch or three at Jet's Bar
to loosen you up for the puddle jump.

It is impossible because there is no runway that would accommodate a jet.

Depending on the time of year you'll be going, you should probably make a
reservation even if you don't book/pay now.  It used to be quite a bit less
busy and one could just wander up to the desk and get on the next flight.
Nowadays, even with a pre-paid reservation, I've had to wait.  Thanksgiving
weekend in 2006 was an absolute nightmare.  Well, maybe not a nightmare
exactly, but once I'm in Belize, I don't particularly want to sit around in
the airport for 2 1/2 hours waiting for a flight to the island, which is
what happened to me and many others.   Even this past February, my scheduled
flight didn't leave until half an hour later.  I'm a bit confused as to how
it works.  There were a number of flights, all booked for exactly the same
departure, but the passengers were given boarding passes with "Departure
Time  3:30 C"  Or "A" or "B".  Near as I could tell "A" left at 3:30, "B"
sometime later and my flight "C", what closer to 4:00.  This is the second
time this has happened to me in the past 12 months, so apparently this is
the way things are run now.  That said, I fly Tropic . . but I'm tempted to
give Maya a try to see how they are handling things.

When I make my hotel reservation, my hotel asks me if I want them to book my
flight.  I always say yes, and they just include it into my hotel bill.

That said, I have handled the flight booking on my own in the past, when not
staying at that hotel.  All you have to do is get online with Tropic or
Maya, and make yourself a reservation.

You could take a water taxi if you wanted to and some people seem to really
like that.  It's not a trip I generally recommend, unless it is to a
backpacker type.