Hello,
to 2): (I don' know how to "whiten" the Latex formulae, therefore I post my reply completely visible)
A right triangle with hypotenuse c and legs a and b has integer sides if
a = uČ  vČ, u > v and u, v in IN
b = 2uv
c = uČ + vČ
(for instance u = 2, v = 1 will give the famous 3, 4, 5 triangle)
The area of a right triangle is
Plug in the above mentioned terms for a and b:
Now you have to prove that

is only a square if u=v or u=1 and v=0

is only a square if u = v

. Solve this equation for u and you'll get:
that means u is not in IN
Thus the area can't be a square.
EB