Fujifilm X-A1 or Sony NEX 5R/T
"photographing kids in the family"
buy Sony (simply because of AF)
You should also consider e-pl5 (superb AF, in-body stabilization)
Image quality is the same (small prints, web)
If the X-A1 AF system is anything like the Fujifilm X-M1 (and it's likely 100% identical), it's quite fast, more than fast enough to capture kids playing in the yard or doing some kind of athletics.
Fujifilm X-M1 AF Demo
The X-A1 comes with a far better kit lens than the Sony 5T. The Sony doesn't even ship with the 18-55 kit anymore, but it includes the absolutely terrible 16-50 PZ. I've used this lens back when I took a chance in the Sony NEX 3N. Needless to say I lasted 4 weeks before I sold both the camera and lens on Ebay for a loss. I would not recommend any kit that featured the Sony 16-50 PZ. If you shoot uncorrected RAW, the distortion is a whopping 8% at 16mm, almost a fisheye.
And in general, the overall lens lineup for Fujifilm is far stronger especially since most of the decent Sony lenses are astronomically priced Zeiss branded lenses.
And more superficially, the X-A1 is a far nicer looking camera.
That said, there are only a couple of reasons to prefer the Sony 5T:
- if you need an eye level VF, the 5T accepts an add-on EVF.
- if you are interested in video, the Sony tends to do video very well, although I haven't tested the 5T to see if it has some of the moire / aliasing issues of some of the other NEX cameras. The 3N I had was not a good video camera at all because of extremely strong moire and aliasing.
So overall, the X-A1 kit is really hard to beat for the price. It's anywhere from $100-150 less than the 5T kit, and it comes with the excellent Fujinon 16-50. And 35 f/1.4 which you mention is just an amazing lens, to me reason enough to buy into the Fujifilm X system.
Best of luck, Markus