I often shoot with a .7 Canon wide angle converter on a Sony FX1 videocam with a 12x zoom.The wide angle converter really works well for most shots.

For an FS700 I need a really good wide angle autofocus that does really well in very low light. I don't think any of the E mount or A mounts do as good in really low light as an E or A mount prime does.

Do I really need an F1.4 or F1.8 wide angle lens for very low light evening shots where I have to avoid noise and grain at all costs?

What very wide angle auto lenses would you recommend for very low light shooting, and and what versatile auto lens would you recommend for everything else?

This site has excellent reviews of all kinds of lenses both for Nikon, Canon and many others, they work well on the 700, with adapters, there are many other solutions as well.
http://www.the-digital-picture.com/Reviews/Nikon-Lens-Reviews.aspx
http://www.the-digital-picture.com/Reviews/Canon-Lens-Reviews.aspx

This is a good solution
Tokina 11-16mm f/2.8 AT-X Pro DX Lens Review (http://www.the-digital-picture.com/Reviews/Tokina-11-16mm-f-2.8-AT-X-Pro-DX-Lens-Review.aspx)

This is the best in my opinion:
Canon EF 14mm f/2.8 L II USM Lens Review (http://www.the-digital-picture.com/Reviews/Canon-EF-14mm-f-2.8-L-II-USM-Lens-Review.aspx)

I have the Rokinon 8mm as well it is 3.5 but it is very good with minimal distortion for a lens that wide.

By the way these will not auto focus on the 700, but with these focal lengths you don't need auto focus IMHO. Personally I don't use autofocus much even on long lenses. I find the 700 with its 4X and 8X magnification and peaking is very easy to focus.

Rokinon has a new 8mm F2.8 for NEX cameras, available currently for preorder.

Available now:

Amazon.com: Rokinon 8mm F2.8 Ultra-Wide Fisheye Lens for Sony E-mount and Sony NEX Cameras 28FE8MBK-SE Black: Camera & Photo

I have no idea how good this lens is, but I was impressed by the 8mm 3.8 with EF mount, for the money.

Is an F2.8 lens going to be fast enough to shoot outdoors at night with dinner lamps and candles, or indoors with poor lighting?

Is an F2.8 lens going to be fast enough to shoot outdoors at night with dinner lamps and candles, or indoors with poor lighting?

That is up to you to decide. What amount of gain can you accept.

Just do some tests with a lens you have now. If you don't have a 700 yet, you may be able to find some footage o the web with exposure info.

I may do some low light shooting in the next few days with the 700. If I do some that matches your criteria I will post to Vimeo and put a link here if I have the right to show the footage.

Is there a reason why the lighting is so low/poor?

f/2.8 and larger should be considered a special effect shot. Shallow DOF looks good when used effectively but gets really distracting when trying to watch an hour of it. Just because you can doesn't mean you should.

Personally I recommend adding enough light to get you up to f/4 or better yet f/5.6. Focus management becomes much easier and backgrounds are still pleasingly soft without DOF becoming so shallow to be distracting.

It is not possible to add lighting while on location in remote parts of the world when it must be carried in checked bags, plus the lighting never looks natural, it looks like someone added fill lights and setting up lights for each shot is too daunting.

Plus existing light always looks natural to the eye. There might be shots of a couple in the evening having a candlelight dinner with ambient evening light, or a couple watching a sunset, or a group of people at an evening dinner table with just small perimeter light bulbs for guests.

It just requires I resolve things with a reasonably fast lens as I have to avoid grain and noise at all costs.

The Sigma 20mm f1.8 is an excellent and fast lens. And it also covers full frame. It's not as wide as those 11mm variants but it also doesn't look like a special effect. And then there's the Samyang 24mm f1.4. Not as wide but a bit more light.

Is that Sigma an auto focus lens?

The Sigma 20mm f1.8 lens is available in a variety of mounts, but NOT in native E mount for NEX. As such, you would have to get the A mount version, and use it with the Sony LA-EA2 adapter. The LA-EA2 adapter allows auto-focus on the NEX FSx00 cameras, however it apparently forces an aperture of f3.5 of any attached lens when auto-focus is activated - at least, this is what several reviews have stated. Seems horribly limiting.

Sigma released 19mm and 30mm f2.8 lenses for E mount with auto-focus earlier this year.

Also, if you are not in a huge rush, Sony indicated in an E mount lens "roadmap" back in January that there would be some additional E mount lenses this year. Since this is a Photokina year, it's certainly possibly they will announce / show some new E mount lenses at this major international trade show in September...