解：（1）$S$的值与$a$的大小无关
理由：由题意，得$S = a^{2}+b^{2}-\frac{1}{2}(a + b)\cdot a-\frac{1}{2}(a - b)\cdot a-\frac{1}{2}b^{2}=\frac{1}{2}b^{2}，$$\therefore S$的值与$a$的大小无关.
（2）$\because a + b = 10，$$ab = 21，$
$\therefore S=\frac{1}{2}a^{2}+b^{2}-\frac{1}{2}(a + b)\cdot b=\frac{1}{2}a^{2}+\frac{1}{2}b^{2}-\frac{1}{2}ab=\frac{1}{2}(a + b)^{2}-\frac{3}{2}ab=\frac{1}{2}\times10^{2}-\frac{3}{2}\times21 = 50 - 31.5 = 18.5$
（3）$\because S=\frac{1}{2}(a - b)\cdot a+\frac{1}{2}(a - b)\cdot b=\frac{1}{2}(a - b)(a + b)，$
$\therefore S^{2}=\frac{1}{4}(a - b)^{2}(a + b)^{2}.$
$\because a - b = 2，$$\therefore (a - b)^{2}=a^{2}-2ab + b^{2}=4.$
$\because a^{2}+b^{2}=7，$$\therefore 2ab = 3.$
$\therefore (a + b)^{2}=a^{2}+2ab + b^{2}=10.$
$\therefore S^{2}=\frac{1}{4}\times4\times10 = 10$
