State of the Market for sub-$400 and sub-16 oz?

Im in the same situation although I have no price constraints.
there are two to consider that produce high quality photos.
The Panasonic Lumix FZ200 and the Canon SX50HS.
The Canon is about 19 oz and the Lumix about 18 oz, a tad over your weight but not by much.
When you mention the NEX consider that it is light only with a light lens which will not get the telephoto shots, so you will be limited unless you decide on packing two lenses, in which case you will be over both your weight and price limits by far.

The Lumix would be my choice (it is in fact what I will get) and the price is dropping by about $15 weekly on Amazon. Right now it's $499. I will pull the trigger when it gets to $450.
The Canon is within your price specs and has the best zoom range bar none, and it gets good reviews.
http://www.amazon.com/Canon-PowerShot-Digital-Camera-2-8-Inch/dp/B009B0MZ1M/ref=pd_cp_p_3
The Lumix, however, is a F2.8 throughout the zoom range and has near twice the battery life and takes fantastic quality shots.
http://www.amazon.com/Panasonic-DMC-FZ200-Digital-Camera-Optical/dp/B008MB6ZX0

Check out the review of the Lumix here:
http://www.dpreview.com/reviews/panasonic-lumix-dmc-fz200

If you don't want to spend for the FZ200, the FZ 70 is similar and within your budget, but ….21 oz.

A good friend of mine has the RX100 and tucks it away in his waist pack for runs and slips it into his hipbelt on hikes. The image quality he achieves is mind blowing from such a little shooter. It has replaced his DSLR for most of his trips now.

I don't think there is anything close to the quality of the RX100 in that price range. I would scrape up a little more shiney and go with the RX100. Amazon has a few under $500 that are used with limited warranty, well worth it IMO.

A few other options you might find around the $400 mark that are capable shooters and backpack worthy:

Fuji X20 (*if you're ok going used, they come up for sale)- large sensor, great build, fast lens…

Panasonic Lumix GX1

Panasonic DMC LX7

If youd like a viewfinder (electronic) then id recommend the Panny g5. Think if seen it in low 400s with 2 kit lens.

Otherwise I think the gx1 is the best deal going on interchangeable lens cameras.

I have the g5 and am very happy with handling/features/quality esp for the price.

Edit: g5 body only is 14oz; gx1 is 10 oz

You might want to take a look at the Canon EOS M. It has the same sensor as Canon's APS-C DSLRs, weighs right about 16 oz with the kit lens, and the kit is selling for $345.

To the OP : Perhaps you could tell us what zoom levels you need. If you need only wide angle then the Panasonic G5K is well proven for "around" a pound with a 14-42 kit lens.

I own the G2 and two lenses – the 14-42 and a 45-200 tele-zoom. I did the math for weights and decided it was too heavy to carry with both lenses, which is why I recommended the FZ200 "All-in-one" camera.

If you need only wide angle you could start with the G5 and kit lens, then upgrade later to the pancake lens for even lighter weight. Virtually all of the Micro 4/3 brands are good from this aspect.
But if you want telephoto capability you will be toting an extra (heavy) lens along.