a thermostat is not essential.  the engine will run without it, but it
will run cold in the winter, delivering less heat than you may want.
the thermostat should be replaced after you figure-out what you're up
against.

if you blew a hose, you need to understand why.  what caused the high
pressure?  was it just a crappy hose, or was it something worse, like a
head gasket?

first, fill the engine to the brim with water, and have a gallon jug of
water handy.  start the engine, and let it run for a while.  keep the
radiator full by pouring water into the radiator.  are bubbles coming
out of the radiator from the engine?  are they exhaust gasses?  if so,
you probably have a bad head gasket.  next test is a compression test.
it's easy, just get a compression tester, and remove spark plugs
one-by-one.  crank the engine (with the spark coil disconnected so it
can't start) with the compression gauge pressed into the spark plug
hole.  record each result.  if there's one that's more than 20 psi
lower than the others, then you may have a leaky head gasket.

there's another test that uses about 10 psi pressure injected through a
special radiator cap.  you pressurize the radiator, and see how fast it
leaks down.  it should take quite a while.  this test can force water
into the cylinders, so i don't like it.

good luck,
bill