I think you need professional representation for an appeal. You are too involved and too emotional. e.g.
1. The company don't care if you have a child that you only get to see at weekends. That's your problem, not theirs.
2. You're engaging in "he-said,she-said" which is not constructive.
3. Clearly your relationship with the relevant supervisor has broken down to the point where he physically assaulted you.
Ask for an appeal, ask to have representation, say that you didn't fully understand what was happening. Get a union rep or someone to represent you - not necessarily a solicitor.
You may have to apologise for walking off the job, and not working the Sunday. But these are not disciplinary offences, especially if other people also refuse to work the Sunday but have escaped disciplinary sanctions.
You have leverage in that you were physically assaulted by your supervisor at work. Not only is this a criminal offence should you choose to report it to Gardai, the company is obliged to provide you with a safe and secure working environment. If you leave your job due to physical violence or a reasonable fear that you will be assaulted, you have an excellent case for constructive dismissal.
But beware that constructive dismissal can be hard to prove and the onus is on you to prove it.
However if the company fire you, you have a good case for unfair dismissal, as you have an open complaint (against the same supervisor?), were assaulted, were subject to an apparently unfair disciplinary process, and have been certified as having work-related stress.