cooling/ventilation/odor control advice please

I am a first-timer and new to the site (obviously). I have searched a lot about ventilation, cooling, and odor control and feel like I get the basics but I was hoping those with more experience could tell me which setup would work best for the constraints I have to work with.

I'm working with a 6'x6'x8' closet located off an unused bedroom in my apartment which is 10'x12'x8'. Although it's my first time, I decided to do my research and be prepared, so I am going to start off with seeds I got from the Doctor. I know it's not recommended to waste money on plants that might not make it, but I really just wanted to see if the seeds would actually be delivered and I also chose some of the cheapest - White Lady ~$35 US.

I've assembled most everything I think I'll need except a ventilation solution. The items in bold are actually from a kit I got on ebay. I know - . I only have 10 seeds so I am transplanting the seedlings straight into the two gallon grow bags. Please let me know if you think of something I have left out or if something I have doesn't really work:
-CFL bulbs totaling 120 watts, mix of 5500K and 6500K: for seedlings
-digital soil hygrometer
-digital soil ph meter
-ph water test kit with ph up/down
-chlorine elimination drops for water

Okay, so back to my original question. I know keeping the door to the closet always closed and cutting holes for intake in the bottom and exhaust via carbon scrubber in the top would be best for odor containment but not for heat control. But if I set my AC low to keep the adjacent bedroom(source of intake air) cool and have oscillating fans inside the closet would my garden still overheat? Or should I get a standalone scrubber and leave the door open to the adjacent bedroom when the lights are on to allow for better circulation. This is where I need someone's personal experience. I would rather keep the bedroom door closed and closet door open but would my whole apartment reek? Sorry for such a long question, any help is greatly appreciated

Whoa. 35$ for seeds? r/d money well spent (unless you have a brown thumb).

OK, if it where me:

Think cool air low,low air in.
hot air high,high air out.
Now, where is the air going to be coming from and going to? The bedroom or closet seems to me from your post (the closet seems almost as big as the room, nice).
All that air recirculating around in a closet might get pretty HOT.
Close room and open closet with air scrubber.
Other than that if you can't find an open window or a bathroom with a exaust, fan see top of reply.
The problem I see is you need a direct line to a window or outdoors, I'm a real big fan of fresh air flow (not the 80's band).
People hear of computer case (12v with 12v converter, all dumster dived) fans and laugh. Then they see and feel the air flow. Quote from the Budda in LB "Well I guess you don't have to worry about air flow, Oi vey." Not that smaller, quieter, more efficient and cooler fans are a replacement for fresh air.

Good Luck

Awesome help, thanks! +rep

I think my issue is really whether to rig up the closet door as you described, with air scrubber inside the closet and intake and exhaust holes through the door, or just leave the closet door open all the time and have the scrubber in the larger room. The bedroom door will always be closed regardless. I just have no first-hand idea as to the actual effectiveness of carbon filters so I don't know whether it would keep the smell inside the bedroom/closet if the closet door were left open when the lights are on. I'm hoping someone who has had a similar setup before could steer me in the right direction to control both odor and heat. My windows are kind of close to a sidewalk so I can't have ventilation ducts sticking out.