Hot glue is an adhesive that is solid at room temperature but liquid when heated.  It usually comes in sticks which you put in a hot glue gun.  The hot glue gun runs on electricity.  The sticks usually run around $3-4 (US dollars) for a bag of 100 and glue guns usually cost around $5-10 and often come with some glue sticks.    I don't consider that terribly expensive, but your opinion may vary.  If you're planning on doing a lot of art projects, a hot glue gun is a good investment; you can also use it for various small home repairs as well.

There are certainly other ways you could attach the limbs of your rabbit, but hot glue probably is the fastest and easiest.  I'd advise applying some strips on top of the hot glue for added strength.  If you don't want to use hot glue, just use regular white glue--you'll have to set up your project in such a way that it won't be moved until the glue sets, which will probably take several hours at least.  I'd probably glue one limb at at time, propping things up with some kind of support to keep the limb in place until the glue sets and hardens.

A hot glue gun is really useful, for all sorts of things - and not at all expensive.  A problem is that it hardens really quickly so you have to work really fast.
To join cardboard, I trail some PVA along the edge, leaving three or four small gaps.  I put a dab of hot glue in each space, press the pieces together.   The hot glue fixes them in place really quickly, leaving the PVA to slowly set and make the joint really solid.
Incidentally, I would glue the cardboard before doing any paper layering.  The edges of the cardboard can be sanded or even cut to make the whole thing smoother before laminating.
DavidO