One thing to consider is that soil (at least when damp enough for plants to grow) is HEAVY. I would be concerned about the strength of the sides of that container, esp. since it seems to be somewhat ripped. If you're going to use the ripped one - or even an intact one like it - I would really suggest constructing a wooden box that it sits snugly inside, with the box sides as high as the container's. This will also help keep the soil cooler as well, a good thing in a warm climate. If you do that, the container should work well!
In addition to drainage holes, put down a couple inches of something very freedraining -- not chicks though

. Use 1" gravel, styrofoam packing peanuts, etc. Ideally then cover it with a layer of old fiberglas windowscreen (can be ripped or holey). Without a drainage layer across the whole container, roots can still waterlog even if you have a good # of holes.
I would not bither sectioning it off, that will reduce the root room available to each plant (plants *can* often 'share'

)
carrots, cucumbers, potatos, romain, spinach, onionsand some herbs: garlic, rosemary, basil, oregano and thyme.
That is probably a bit much for just one container

but if you do multiple containers, you're set. I'd probably put potatoes and spinach together, with lettuce as space permits; cucumbers in their own smaller tub with early lettuce around the edges (possibly onions too? I dunno about container-growing onions); and then the herbs in their own smaller container because they need drier faster-draining conditions and would also be in danger of being grown over by many of the other larger things.
Set up an auto watering system, even just a big jug you refill daily that bleeds water into the pot all day (in addition to daily watering)... containers go thru a LOT of the ole H2O and everything you name except the herbs will be adversely affected by letting go dry.
Have fun,
Pat