Don't include the email with your appeal REQUEST. That's all it is. Just write, "the determination dated MM/DD/YY is wrong. I want an appeal hearing scheduled."
Save the evidence for the hearing.
Also, it doesn't matter how great an employee you were or for how long. Get that idea out of your head.
It all comes down to is can the employer prove that you took a day off, that you didn't have permission, and that doing that one time for one day rises to the level of misconduct. I say no. You simply had a bad adjudicator that was intently listening to the employer to cherry pick a disqualifying reason, and hoping you won't fight it, or are so desperate for cash that you take the first mcjob that comes your way.
Stay focused. You don't have to try to prove everything out of the employer's mouth is a lie. Being able to substantiate that you had permission is sufficient. Heck, this would probably fall under the one-time, isolated incident rule, or it also sounds like you were never given a warning to never do it again.
You can submit the email about your personal life at the hearing. If nothing else, it suggests that your employer has no clue why they terminated you, and it makes them sound like they are making stuff up on the fly. However, keep in mind that the employer might not have said things the way that your decision reads, but that the adjudicator did that.
You'll know more after you get your hearing packet. You'll be able to view the file and the deputy notes prior to the hearing if it's in-person, or everything will get mailed to you if it's a telephonic hearing.