Algorithms for set packing
Indeed, set packing, partitioning, and covering have been studied in terms of exact algorithm running times.
 To address your last question, you can solve weighted set packing in $O(m2^n)$ time by dynamic programming across all subsets of $[n]$.
 Moreover, if your integer weights are bounded by $M$, you can solve it in $O(M2^n)$ time, even if $m$ is as large as $2^n$, see

http://dx.doi.org/10.1137/070683933

BTW, The parameterized result you list for $3$-sets is not the best known, see

http://arxiv.org/abs/1007.1161

for a state-of-the-art algorithm and a list of previous results on the problem.