What others get away with won't be relevant to your case.
key thing is honesty - its a slam dunk for an employer to dismiss if you lie and can be shown to have lied.
Apologise for the short comings, state how much you like working there, need the job etc. - state it was a family emergency and that you had to deal with it, it took longer than you thought to sort out. You didn't report it because you were worried about getting in to trouble. (its better than you thought you could get away with it)
keep your mitigation as short as possible - the HR dogs will have been trained in all this.
If they then dismiss you or you think the punishment is overly severe, then you appeal on the grounds that you haven't been treated fairly and cite other cases.
If you have legal insurance with your credit card, home insurance etc.. give them a call.
If you want to go and they want you to go, then a compromise agreement might be the best option (they pay you to resign).
I know right now.. it seems like leaving and starting your own business is the right option, but it absolutely isn't. The best way to do this, is to secure your job and plan the whole thing while you still have an income. Most people who've never set up a start up focus on the logistics of the work. Its not the place to start. Start small (weekends, downtime) and build. 10 x more likely to take off. I've been a business angel in the past and those that make it don't start off from zero with big costs.
Best of luck