I am in an apartment also. You need to grow in a contained area. You can use the closet if you intend to dedicate the closet solely to growing. Or you need to seperate your growing area from other stuff. There are tons of little issues along the way. I PROMISE.

Venting CAN be simple. Do not buy an ionizer. Buy a carbon air filter canister. They are about $75 and you will have to vent the area through that. With an inexpensive fan... You can buy them online or at most local grow suppliers...

The carpet WILL be an issue. I would just cover it with a tarp to keep bug problems down. Then just TAPE all the way around the perimeter to keep it sealed.

THE MAIN THING IS: Be able to move your grow in FIFTEEN minutes. NO EVIDENCE LEFT BEHIND. No holes in walls, etc!!!