To your first question, yes, the 500 texts are domestic. That package also comes with unlimited texting to other Verizon Wireless subscribers.
To your second question, the account owner can view the numbers and date/time of any call or text sent or recieved on any of the lines on their account. They can do this online, or in a detailed paper bill. If they don't have paperless billing, and they've chosen to recieve the basic bill, then the full call/text detail won't be available, but if they ask for it, they can get it.
In short, the account owner will be able to find out who you are texting/calling if they want to. If you want privacy for those records, you should consider moving your line to your own account.