Employment Law Questions? Ask an Employment Lawyer.
Yes, the employer is not required to give you the evidence against you, because they are not a criminal law court and you have no legal right to confrontation of the witnesses or evidence against you (as you would in criminal court).
If the email is on their computer system, they can access it and use that against you.
Resumes can be considered confidential information by the employer. There isn't a federal statute making them confidential, but if the employer doesn't wish for them to be released that can be a terminable offenses.
In terms of legal right, unless you had a contract of employment stating that you could only be terminated for cause, your employment is "at will" and can legally be terminated at any time, with or without cause. Unless you can allege the employer here had a different motivation, such as your race, religion, gender, age, disability or recent FMLA use, the termination would be legal.
As for proving why you were actually terminated, it doesn't actually legally matter unless you can prove that the reason they actually terminated you was illegal. That's your burden. Even if the reason they gave is not true, that doesn't matter. It doesn't have to be true.
Yes, they can use your personal email, if located on their system.
It does not matter that they haven't stated previously that resumes are confidential. They can just personally believe that they are confidential. Again, without a contract of employment, they don't need a good or even valid reason to terminate you. They could literally terminate you for absolutely no reason and that would be legal. They could terminate you for liking the color blue too much and that would be legal. They don't need to have had an issue with your performance or to have written you up before.
Now, you mention retaliation about your complaints concerning unfair pay practices. If you made a complaint about discrimination in wages due to race, religion, gender, age or disability, then yes that would have been an illegal basis for termination and you could file a complaint with the Equal Employment Opportunity Commission.
The retaliation that you appear to be mentioning here is not related to a complaint that you made to the employer concerning race, religion, gender, age, disability or FMLA discrimination against yourself. Therefore, it would not be an illegal form of retaliation.
You also continue to state that there was no written policy with the employer stating that the resumes were confidential. I have already said before that is legally irrelevant. Unless you had a contract of employment that specifically and unequivocally stated that you could ONLY be terminated for cause, they need absolutely no reason to terminate you. All of your additional facts, while certainly indicative of an unprofessional work situation, DOES NOT change the fact that you are an "at will" employee and can legally be terminated without cause.
I know this is not what you want to hear, but you have stated no legal basis for a challenge to the termination.