Quick summary: if you use the 24-70II, you will want to use a flash. You could bump the ISO a couple of stops but this seriously degrades image quality and defeats the purpose of using a lens like the 24-70 -- you're basically using 5-6k worth of gear to get inferior results to a Rebel with a 50mm f/1.8 and a flash.
I would recommend considering a flash even if you do get a prime lens. If you don't use a flash, shoot raw for nighttime shots (you will get better results if you tweak the white balance of the raw files). Especially for nighttime shots, it will provide more natural (less yellow) lighting, and provide you with an additional light source that you have direct control over.
As others have pointed out, there are some cases where a flash really helps (you need more depth of field, or you are trying to stop action). For best results, you want all options on the table (fast glass and a flash).
1. Did you find the current EF 24-70L f/2.8 (set ~50mm) fast enough for this?
No, for stills I often want a faster lens indoors (unless I'm using a flash). For video on the other hand f/1.4 is too shallow dof for anything moving.
2. Did you find the need for a (say) EF 50 f/1.4 necessary?
You could get by with a slower lens but it's not optimal.
3. Is increasing the ISO (now possible on 5DMk3) a workable option. Why not?
I have the 5DII, not the 5DIII. Even at f/1.4 it wants to step all the way up to ISO 3200 for indoor shots (there's a 6400 setting that I never use)
The 5DIII will let you increase the ISO further, but at the expense of degraded performance. Even though ISO goes 4 stops higher than the 5DIII, image quality isn't 4 stops better.
4. Is opting for a Zeiss ZE 50 f/1.4 and resorting to MF/practice a workable option (to the EF 50 f/1.4)?
For video -- you will need to manual focus either way, so a manual focus lens is probably better (focus ring will have a longer throw which will make it easier to manually focus)
For stills -- I've used my EF mount lens wide open manual focus on a micro 4/3 camera to take pictures of a toddler. You will get a lot of missed shots, but you don't need a 100% hit rate. You have something that pros have very little of -- time with your subject. It really depends on whether you enjoy the challenge or find it frustrating.
If you do use a manual focus lens, you will probably want to replace the focusing screen (something which can't be done yet) -- the viewfinder will give you f/2.8 depth of field, leaving you dependent on the focus confirm beep to determine that you're in focus if you're shooting wide open.