Just wondering if when I submit my application they can check what corp the other characters in my account are in as well in addition to the corp history of my applying character.
no. They can only see the ESI of the particular pilot. They cannot see what else is on the account.
so I can join rival corps with different characters on the same account? Are there any drawbacks to doing this?
no. unless you need both online at the same time, which you cannot do. but as far as applying characters, they can see corp history in game. or through 3rd party sites like evewho.