FlyGuy, what I'm hearing from you SHOULD be nothing but a sympathetic situation. Having family issues during airline training is rough. My mom had some medical issues when I was upgrading at my previous regional, so being forced to push that aside and deal is a monumental task. I commend you for attempting to do your best to do so.
With that being said, I zoned out during your explanation pretty quick. I'm pretty confused. You seem to be willing to take the blame while also faulting those in the training department. I have found in my years of airline experience (especially regionals) that the training department is untouchable in terms of standards, professionalism, etc. You're up against a buzzsaw and it's simply up to you to get out of it. Unfortunately, you didn't.
Personally, I'm not sure you going to Horizon and explaining the situation would have done a lot of good. Some airlines would be sympathetic; others would tell you to suck it up and deal. Sounds to me like Horizon would have just said to deal with it. Right or wrong, the 121 world is cutthroat. Always has been and always will be.
In your situation now, if you think you have learned your lessons and can power through, I'd keep applying. Something will stick here soon.