2. What is $t$? a constant? if it is then $F(f)= 1+xf(x-t)$ which is clearly a contraction. The fixed point of this function however is not so easy to find (to me at least).
If however you meant $\int_{0}^{x} f(x-t)dt$ then, arguing as in your last question, one gets it's indeed a contraction and using induction we prove that $F^n(1)= \sum_{j=0}^{n} \frac{x^j}{j!} \rightarrow e^x$ (where $F^n$ means composition $n$-times)