So your bank account is getting mixed up with your cousin's; your bank is closing out accounts without permission; your dad has a habit of stopping checks; and your mom is resetting your statements.
You say BofA doesn't have a branch within 500 miles of you? You're in Kansas, correct
? There are 50 branches or ATMs in Topeka. There are 40 branches or ATMs in Wichita. Hays, Dodge City, Great Bend... they all have branches and ATMs.
If you have a smartphone, Bank of America supports remote deposits (i.e. you take a picture of the front and back of the check and upload it.) I never deposit checks any other way these days.
But (apart from the massive family issues that need resolving of course!) I would suggest a true internet bank, such as Ally, Everbank etc.
Most of these allow you to withdraw from any ATM and refund all fees charged every month. Most have very low starting deposits. Now these won't help with credit cards particularly (well, Everbank have their own) but getting financial privacy is a critical first step. Your parents won't be able to convince a CSR on the internet to stop checks, show statements etc, providing you can resist giving them your passwords.
Instead of the TV, consider using the money to consult a therapist (in a different town). It sounds like you are in a very toxic family relationship and after all these years it's hard to change, and someone else's point of view would be very helpful. As you can tell from most of the responses, this situation is far from normal, and most cannot even begin to understand how this can happen.