$a)$

${\left(-3\right)}^2=\left(-3\right)\cdot \left(-3\right)=9$  

$\left(\frac{6^2}{2}\right)=\frac{6\cdot 6}{2}=\frac{36}{2}=18$  

${\left(-3\right)}^2<\left(\frac{6^2}{2}\right)$

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$b)$

$3^4-2^4=3\cdot 3\cdot 3\cdot 3-2\cdot 2\cdot 2\cdot 2=81-16=65$

${\left(-6\right)}^0+4^3=1+4\cdot 4\cdot 4=1+64=65$

$3^4-2^4={\left(-6\right)}^0+4^3$  

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$c)$

$-4\cdot {\left(\frac{1}{2}\right)}^2=-4\cdot \frac{1}{2}\cdot \frac{1}{2}=-1$        

$-{\left|-0,3\right|}^0=-1$  

$-4\cdot {\left(\frac{1}{2}\right)}^2=-{\left|-0,3\right|}^0=-1$