I think I solved (i) by using the Fisher-Neyman theorem of factorization and I separated the joint likelihood into two functions, such that one, I called it h(x), does not depend on theta, only includes x, and thus I got a sufficient statistic.
I am stuck on (ii) because I don't quite understand the question and how I'm supposed to transform my answer from (i) to Bernoulli, Poisson, and Exponential. Any help is appreciated!
Of the three, the exponential is the simplest
$$f(x_1,x_2,\ldots,x_n; \lambda) = \lambda e^{- \lambda x_1}\mathbb 1(x_1 \in \mathbb R_{\ge0}) \lambda e^{- \lambda x_2}\mathbb 1(x_2 \in \mathbb R_{\ge0}) \cdots\lambda e^{- \lambda x_n} \mathbb 1(x_n \in \mathbb R_{\ge0}) \\= \lambda^n \exp\left(\sum\limits_1^n -\lambda x_i\right)\mathbb 1(x_1,x_2, \ldots,x_n \in \mathbb R_{\ge0}) \\= \exp\left(-\lambda \left(\sum\limits_1^n x_i\right) +n\log_e(\lambda)\right)\mathbb 1(x_1,x_2, \ldots,x_n \in \mathbb R_{\ge0})$$
which is of the desired form.
The other two are similar and you should try to do them yourself