First - YOU own all the rights to the code, since you created it - even if they paid you. If you sign that contract, THEY own all the rights to the code.
Second - It is up to you how you want to proceed, but if you want to own the rights to the code, just keep working without a contract.
Third - DON'T involve any other lawyers or write up your own contract or call the school's tech transfer office. This is only going to set up the situation to favor the other guy or the school. Remember, YOU own everything.
Fourth - The lawyer dad knows all of this and that is why he wants you to sign the contract. Lawyers are assholes - DON'T let him bully you into signing anything.