I understand that the carbon filter lasts, if I se...

I tried like 3 cheap brands, all lasted 12 month and were changed only because of beeing too cautious. Infinity lasted 10 month, most expensive i tried by $. I always had one spare, so keeping/storing it for 6-12 month didnt changed how they lasted during real work...

Very slowly.. it can interact with only so many molecules before it is no longer effective, but not much air is interacting with it if there's no airflow.

Yes, if I'm understanding you. Air running through it. Which in a tent should be 24/7 after at least 2 weeks. I use 24/7. Believe it or not. The smell of organic nutrients & soil smell strong at the start. And anyone who knows weed or gardening will smell it. As they will with some filters (that's if they know filter smell). Less do, so. That smell goes, but if you don't mind the gardening smell. Then you can use it a few weeks into your grow to cover the cannabis smell. But, 24/7 is best. They last typical 4/6 months. Mine do. And i still have old ones. And after a few months of been left in my shed I get another few weeks run, so I rotate about 6 every year. You can also get refill ones and just buy the charcoal. Or buy decent prized ones and they'll last you at bare minium a full 3-4 month tent grow.

Only with the extractor on will the air (and particles) go through the carbon filter so no, without the extractor the filter will be just fine.