If I have deranged hermit on the battlefield, and someone casts a rain of embers, will the squirrels die due to the 1 damage that is marked on them after the hermit goes to the graveyard?
Yes. The first check of SBAs after the Rain has resolved will destroy the Hermit. Then, since something had to be done by SBAs, another check is done, now seeing the 4 1/1 squirrels with lethal damage marked, getting them destroyed. A third check then seeds that everything is fine and the game puts triggers on the stack, then gives priority to the active player.
Note, that the Hermit and the Squirrels don't die at the same time, which can matter for certain triggers.
Yes; in fact they will be destroyed (assuming you mean the Squirrel tokens created with Deranged Hermit).
After Rain of Embers resolves, Deranged Hermit is destroyed as a state-based action because it has lethal damage (C.R. 117.3b, 117.5, 704.3, 704.5g). After Deranged Hermit leaves the battlefield it will no longer give any Squirrel creatures +1/+1 (C.R. 113.6, 611.3b), so the damage marked on the now 1/1 Squirrel tokens is now lethal damage so they too are destroyed by another round of state-based actions (C.R. 704.3, 704.5g).
Note, that the Hermit and the Squirrels don't die at the same time, which can matter for certain triggers.
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After Rain of Embers resolves, Deranged Hermit is destroyed as a state-based action because it has lethal damage (C.R. 117.3b, 117.5, 704.3, 704.5g). After Deranged Hermit leaves the battlefield it will no longer give any Squirrel creatures +1/+1 (C.R. 113.6, 611.3b), so the damage marked on the now 1/1 Squirrel tokens is now lethal damage so they too are destroyed by another round of state-based actions (C.R. 704.3, 704.5g).