You asked "Am i allowed to let someone else use my iTunes account?"
Apple says you shouldn't:
"You shouldn't share your Apple ID account information with other people. Each person should have their own Apple ID."
Note that they don't say you can't… and they do very little to stop you. Apple's DRM is fairly "light" and meant to keep honest people honest. Purchases aren't tied to a single device (I think you can install on up to 10) and can be re-downloaded at any time.
If developers find more and more people sharing their AppleIDs, they may push for more restrictive DRM in the future. Each actual lost sale takes money out of a developer's pocket: a developer who may have kids of his/her own to support.
If you're asking about the risk of Apple finding out, that seems fairly unlikely. What would they do if they found out? No one knows for sure, but they could terminate your account meaning that you'd have to re-purchase all the apps and music that you had purchased.
Others might say that you aren't doing anything except sharing and while sharing with 1,000 people might be wrong, sharing with 1 is probably statistically insignificant. They would say that your cousin doesn't necessarily reflect a lost sale for each of those apps, because they might not have purchased the same apps.
I have friends who are iOS developers. I do my best to support the apps that I use frequently. My wife and mom share my Apple ID for purchases. I use 80% of the apps that I buy (20% are apps that I buy and then never use, because Apple doesn't have demos). They probably use less than 1% (I'm not sure my mother uses anything except WordsWithFriends, email, and Safari). When I decided to set my Mom up with a Simplenote premium account, I paid for a separate one rather than let her use mine. My wife and I have separate MotionX subscriptions. And when my mom's best friend bought an iPad, I set her up with her own account. In my mind, this frees me up to buy more apps to try more apps. Some will see that as nothing more than an attempt to justify my own behavior.
So those are the "risks," such as they are. There are few concrete ones.
At the end of the day, the answer is "you're not supposed to, but you can" and it's up to you to decide what to do with that.
p.s. - the most likely "risk" isn't what Apple would do if they found out, it's the fact that you will almost certainly have to share your Apple ID password with your cousin. That password is needed for installing free apps and updates, but it can also be used for app
purchases and in-app purchases which can run into the thousands.
That same password can be used for purchasing TV shows, movies, etc.
You'd be responsible for those charges. Unless you see your cousin often enough to do these updates yourself, that's the biggest (practical) risk you face.