The A5000 just got announced, it's replacing the NEX 3n. The next two replacements will come in a few months, so no equivalent yet. There will likely be an A6000 or A7000 model being release which will be an updated NEX6. I owned the NEX6 and it certainly doesn't need upgrading but they're always welcomed. I sold it to get the updated but bought an RX1r as well for now...
I use one of the small cases you can hang on your belt, the ones with a velcro-attached cover, not a zipper. Just check the size, depends on which lens you have mounted. The Lowepro Rezo 50 might fit, the 60 will definitely fit. I wear it on my belt and pull out the camera as needed. It's much more convenient for me than the bags with zippered closures.
I think the NEX-6 would be "okay" - maybe even GOOD - for most home videos. It has an excellent sensor and HD quality processor. You can keep the camera pressed to your face as you shoot a video, because the camera has an electronic viewfinder. By contrast, you cannot shoot a video with a standard DSLR looking through the viewfinder - you must hold a DSLR at arm's length to make a video. It's hard to hold a DSLR that way. The image will jiggle and your arms quickly tire.
The problem could come if you plan to edit different videos into a single piece. The NEX-6 lacks an external mic input, which is essential to making good videos. The internal microphones are OK if you are close to your subjects in a quiet room. If you are far from your subject - for example, your child's soccer game - you will hear only the sounds around you on the sidelines. People talking around you will ruin the video.
The NEX-7 has an external mic input, so you could use an inexpensive (or expensive) directional ("shotgun") mic. But the NEX-7 has amateurish AGC (automatic gain control) on the mic input - you cannot set mic sensitivity, and the sound will "pump" (go up and down) when people stop talking.
Frankly, the advantage of the NEX-6 is that it gives you an excellent camera in a very small light package. It is my hands-down favorite for travel. However, a camera with at least a basic adjustable mic input like the Nikon D7000 may be a better video camera - at the cost of a much larger and heavier package.
Optimum may be the new Sony A7 full-frame camera. At less than $2000, you get a camera with an electronic viewfinder and a very professional mic audio system. The fact that is has a full-frame sensor is almost a bonus.
No camera can replace a camcorder yet. Due to film industry licensing limitations, non-camcorders are limited to 30 minutes of recording regardless of memory available. Some cameras are limited even further for technological reasons (heat, small file limits etc...). Camcorders are generally allowed to record for as long as memory/batteries will allow.
Some cameras do not handle heat from the sensor well when it is in constant use, and will over-heat in a matter of minutes.
Other than that, I simply have not used both side by side to compare quality of one v.s. another, and I have not used the NEX in movie mode to any great extent.
If you are looking for recording more than 10-30 minutes at a time, get a dedicated camcorder.
Yes you can do time lapse photography the Sony NEX 6 via a downloadable application. Here is a link to a youtube video showing it in use. https://www.youtube.com/watch?v=tXxt0VfbHDU
There are also solutions available that allow for connection of your phone to the camera and then run an intervalometer on your phone... Hope this all helps. Good luck, Jim