Does the iPhone support TRIM for flash memory?
Since I'm just a generic curious who can do some research on the web, I'll make here some assumptions so we can be on the same grounds:


TRIM is an SSD technology and, as such, can mean too many  different things.
iPhone uses MLC NAND memory, which is indeed a SSD.
We, people, (such as dhy8386 and maybe even Mr. Brian Klug) love to talk big about things we don't know deep enough. To know more, trust less.
Apple mostly does not disclose anything technically meaningful about their technology. And it does use many proprietary technologies nobody else does.
Even today, the iPhone will work way more flawlessly under deep stress (filled memory, many applications, etc) than any Android or Windows Phone. (yes, I've used them all and I currently own a S3)


If you follow every link above deep enough, you may realize it's just not that simple. Even Android's TRIM support probably isn't the same TRIM as a notebook uses on its SSD.

So, it doesn't matter if it supports TRIM or not. The whole point of going with an iPhone is trusting Apple and letting them do the hardcore tech work. And the whole point of jailbreaking is being able to do things Apple can't allow for any reason, such as unblocking carriers. Anything else on JB is doing it wrong (specially now we have android, even for iPhone) - and I'm saying that from my heavy experience using jailbreak from iPhone 3 to 4.

Now, to address your main and unannounced question, best thing you can do to test if your hardware is malfunctioning is to backup your iPhone and reset to factory. Then fill it up with music or pictures, since you suspect a full disk is causing the malfunctioning, and avoid installing  applications at all. You probably won't see any problem and, if you do, just send it to warranty. You can also send it to warranty directly, but you risk of it coming back and just losing few days there.