I like the 14th, but with kids , might choose to stay in the 6th,, you can walk to alot of sights, and Luxembourg Gardens are great for kids. There is absolutely nothing wrong with the 14 th though.
There are many reputable agencies, do a check on this site, I am using Parisbestlodge,, but there are about 5-6 other well reviewed companies here. I prefer ones that do not require a huge up front deposit,, and I am leary of some of the VRBO ,,some have had great expereinces with individual ones, but its the "by owner" that bugs me,,, so if I chose one it would be a specific apartment someone here stayed at and recommended.