I did read your blue suggestions and deliberately commented only on the things that I thought you had missed.
«"I unlocked _the_ dog’s chain", I suggested the leash instead»
Every dog's chain is a leash, but not vice versa. Otherwise "unleashed" is OK. Hope you didn't overlook m correction — the definite article.
«I'm believe the cracking sound came from the soon to fall branch. That fits the context better. Otherwise, why would anyone mention it?»
Yes, but, the narrator didn't know what was happening until after the accident, right! So I think "as if" works well to describe the happenings from the narrators viewpoint.
«"very thin sound as if it _was/were coming_ from far away" Is afar OK or too poetic?»
Yes, as long as you say just "from afar", not "from an afar place", which is incorrect.
«"personal preference, 'a huge tree bracnh'" — this way lost is the emphasis of the branch's belonging to _that_ tree. Do you think the emphasis is needed though? I was worried it would be a redudant emphasis since the passage had mentioned the location of the person)»
Yes, it is a bit redundant, but to me it makes the tale more interconneced. Personally I'd use just "a branch" without mentioning either "the tree" or "a tree" and thus avoiding both redundancy and "weak coupling"!
«"made me very afraid" — terrified me a lot, maybe...the original didn't sound natural to me either but couldn't find anything wrong.»
What's unnatural is wrong! Also try searching "very afraid" and "made me afraid" is Google, to grasp the usage better.
«"Or else I would have _got_ in (the) hospital" or "would be in (the) hospital" (mixed consitional) Is'be' wrong? I have no problem withyour suggestion but think either is fine.»
1. ...or else I would have spent a month in the hospital — Type 3 conditional (hypothetical past event)
2. ... or else I would be in the hospital (now) — Mixed conditional (hypothetical present event)
As to "would have been", it doesn't make much sence to me, but maybe it's me who's wring? What do you think your version means?