The first part of your schematic basically describes a dessicant dryer or deliquescent dryer. These are common industrial air drying processes that produce very low dew point properties. Generally air (or other inert gas) is passed through a column of desiccant, under increased pressure.
Desiccant dryers use common desiccants that include activated alumina, molecular sieves and silica gel. In all cases, there is a maximum amount of water that can be adsorbed/absorbed before the desiccant needs to be regenerated; typically regeneration is done by heating the desiccant column, although other methods, such as purging can be used. Industrial desiccant dryers have the heater element installed as part of the construction, although any bench chemist can probably tell you about heating red silica gel beads from desiccators in the oven until they turn blue.
Deliquescent dryers are similar, but involve a highly hygroscopic agent that slowly dissolves, or deliquesces, as it absorbs more water. Usually the agent cannot be regenerated, and the waste liquid is drained and discarded. Fresh drying agent then needs to be added. This is most likely the scenario for NaOH.
So, this would need to be a major consideration for your proposed system; how to safely and effectively dry your NaOH. This is a very caustic material to have to try to continually regenerate or replace.
Yes, I would be concerned that you would transfer NaOH from the first chamber. To check this, you could easily test the gas outlet with some litmus.
Can you use dry air to remove excess water from other materials (such as acids)? In theory, this may be possible and will depend on the relative hygroscopicity of the drying agent and the material to be dried; you need to be sure that the air you are supplying your wet material with is dry enough to receive water and not simply supply more. Purge drying (passing dry air over a desiccant) is another form of regenerating industrial desiccants, but is not as effective as heating. Sulfuric acid is itself used as a desiccating agent as it is very hygroscopic. I don't know the relative hygroscopicity of NaOH and H2SO4/CH3COOH, but there is no way I would ever consider passing air from one container directly into the other chamber. This is a very hazardous setup, and is fraught with much danger. Heating an acid solution in a suitably ventilated environment under controlled conditions is still, to my knowledge, the most effective way to increase the concentration of the solution.
You should also consider the energy efficiencies of trying to 'dry' a sulfuric acid solution by passing dry air over or through the solution versus heating it. The amount of time and materials required to produce sufficient dry air to effectively concentrate a sulfuric acid solution is likely to be far in excess than the time/material investment required to simply heating it.