The a6000 is a great camera, however there are a few things that make it expensive...
-Build quality is higher than the a5xxx series and the NEX series <<< it's nice but not necessary
-AF speed is fast and has up to 11fps with continuous focus!! <<< You do not need this
-EVF <<< this is a very nice feature for video!!! (think bright sunny days)
You mentioned the EVF is something you like and I believe at this price point, the a6000 may be the best camera. You can get an accessory EVF for the NEX cameras or you can get a viewfinder for the LCD screen, but this brings the cost equal to the a6000. The sensor on the a6000 is newer than the NEX sensors, so you will have about 1/2 - 2/3 stop better ISO performance. This would make your video a little less noisy. For $200, I would personally go for the a6000. I do believe the a6000 suffers from sensor overheating, however this would be an issue if you plan on shooting long clips (I heard it is fine up to 30 minutes). I would not use this as a main camera to record a once a lifetime event like a wedding ceremony, but I would definitely use this for making a short (or long) movie.
If you want, get the body only (approx $550) without the kit lens and put that $150 savings into a better lens. The kit lens is a good all purpose lens for outdoor, well lit situations, however indoors, you will be very limited unless you use lighting. Also, the variable aperture means every time you zoom in/out you will have to change ISO or alter your lighting. This can waste time or lead to more post work. I think fixed zooms and primes are the way to go for video.
Your best first native lens will be an ~35mm prime. That gives you the 32mm/1.8 Zeiss Touit at $500 and the Sony 35/1.8 at $450. You can pretty much use this focal length for everything as it is approx 50mm Full frame equivalent. The Zeiss is a manual focus only lens and I would go for this one as a native prime personally because of build quality and excellent manual focus feel. The Sony is an excellent lens however and if you want AF, this would be the way to go. Something to consider however is that both of these lenses are APS-C meaning that if you later on purchase an A7 series, it will only work in APS-C crop mode meaning the FF sensor is wasted. You can purchase a Full frame 35mm lens, however this will be more expensive and larger.
Your best choice in my opinion is actually adapting the Sigma 18-35/1.8 (either Nikon or Sony). Again only APS-C, however this lens is a gem!! It is sharp wide open and has a great feel for Manual Focus. The focal range is perfect as it is equivalent to approx 27mm-54mm! This lens replaces two primes and it is well worth the $800 price tag. This will take you over your budget. You can maybe get lucky and have the set for around $1200 with adapter. The would be by far the best you can do for $1000-ish. That set up would be equivalent to something like a 5D mkIII with 24-70/2.8 and about $3k cheaper!!!