The question is regarding SU(2) group and SU(2) algebra. The SU(2) group can be generated by exponentiating the generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$ with $t_a$ being three parameters. Genreally we use half the Pauli matrices as $X_a$ when discussing SU(2) algebra as well as the group. But if we use the spin-1 representation of $X_a$ (which can surely be done at the Lie Alebra level) can they be exponentiated to give another rep. of the SU(2) group (which would be 3 dimensional)?
Yes. A representation of a Lie algebra always exponentiates to a representation of the simply connected Lie group corresponding to it. The representations of each are in bijective correspondence this way.
I recommend the free online book by Kirillov ( http://www.math.sunysb.edu/~kirillov/mat552/liegroups.pdf ) as a reference for this business.