解析这道题---**Question 8:**
**(Context)**
(2024秋·沈阳期末)
**Question Stem:**
已知定义域为R的函数 $f(x) = \frac{a - 2^x}{b + 2^x}$ 是奇函数。
*(Translation: Given that the function $f(x) = \frac{a - 2^x}{b + 2^x}$, whose domain is R, is an odd function.)*
**Sub-questions:**
(1) 求$a$、$b$的值;
*(Translation: Find the values of a and b;)*
(2) 判断$f(x)$的单调性;
*(Translation: Determine the monotonicity of $f(x)$;)*
(3) 若存在$t \in [0, 4]$，使$f(k+t^2) + f(4k-2t^2) < 0$成立，求实数$k$的取值范围。
*(Translation: If there exists $t \in [0, 4]$ such that $f(k+t^2) + f(4k-2t^2) < 0$ holds, find the range of real number $k$.)*