a few flaws that i see..

first off i think that you have too many lights daisy chained together. you are loosing efficiency with the 12" fans being reduced to a 6" duct. air doesnt work that way. you must increase the size of the duct in order for it to be efficient, for example if i was starting with a 6' fan i would increase to 8" or 10" ducting.

next.... the "Y" at the end of you duct is also causing restriction, slowing down the air flow exit.

another factor to consider, is the temp of the starting air before it is heated by the lights, if the air is 100 degrees BEFORE it makes it to the hood. how much heat are you removing?
if you plan on adding an a/c, i would consider oversizing the a/c unit and sealing the whole room. i think that with the 2 12" fans sucking the air from your room will negate the effects of the a/c unit. hope this helps.

Yes you need to redo it! From your pic it seems to me like you should move your intake to the opposite side of your grow, remove the ballasts from the grow, remove the Y connection and have 2 lines of ducting exhausting the air, raise those carbon filters up off the floor also, I'm guessing you're using different size reflectors that's why your ducting changes between lamps? You have 6000 watts running, how big is your room?

are you exausting lights through your carbon filter? If so I would just run one carbon filter scrubbing and hook all those lights to the 12 inch max fan. 3 lights chained together is not too much as i have chained 6 lights to one six inch fan with only a 12k btu portable cooling it (but I live in a cold climate). Also yeah I would make 2 intakes, one for each row of lights as the cooler air will help emmensely. Also, as mentioned previously, remote locating the ballasts will help a lot too, especially if they are magnetics!

You definitely need to replumb your ventilation. If you use two intakes hooked to the lights and a separate fan circuit to ventilate the room your heat problems should come under control.
With two intakes (8in. flanges and ducting), one 12in fan should cool ALL the lights. Use hard ducting wherever possible. As stated above, the Y connector is restricting flow. At any HVAC supply shop you can buy a Y with 2x8in dumping into a 12in. Attach the fan at this point and use 12in duct for exhaust. This should cool your lights very well if you can keep the bends to a minimum (hard duct adjustable elbows).
For room ventilation, if sealing the room and CO2 injection isn't possible, use the other 12in fan to bring in fresh air. As stated, mount the filter up high and use 2x8in. intakes on the opposite side of the room. This draws the fresh air across all the plants before exhaust as well as cooling ALL of the room.
Make sure you tape/seal EVERY joint and connection in the ducting to eliminate smells leaking out. This plan may allow you to do without an AC unit. Maybe. I'd have a small one (12k btu) on hand just in case. Selling the extra filter should cover that cost.
Hope this helps,
Fred