Can I redeem all my iTunes gift cards at once? For the past ever, I've been on my parents' Apple ID, making purchases on their credit cards and sharing any iTunes gift cards I redeemed with the family. However, I recently created my own Apple ID, complete with my own payment method. I'm tempted to upload every single gift card I have to my account. Is this doable, and are there any limitations to doing it?
Yes - stacking gift cards is not an issue. You will just carry a larger balance and less risk of losing the cash value if you lose the redemption codes or physical cards. 

You lose the flexibility to redeem them on another account. So if you’re not the family organizer, don’t redeem a balance or they will need to cut you from the family temporarily when you want to spend your credits. 

Gifts do not redeem twice and you cannot transfer a balance to another Apple ID or get a cash refund.