In CoC 7th edition, you gain 1d10 skill points every time you fail a skill test. Does this also apply to Cthulhu Mythos? And does Cthulhu Mythos get subtracted from your sanity as it did in previous editions?
The standards established in previous editions of Call of Cthulhu (CoC) continue to apply.
When you create a character, you cannot assign points in Cthulhu Mythos as a starting skill (under normal circumstances).
In CoC 7th edition, you gain 1d10 skill points every time you fail a skill test. Does this also apply to Cthulhu Mythos?
You cannot improve Cthulhu Mythos through the standard system of use-based improvement. There is no tick-box for Cthulhu Mythos on the investigator sheet. You cannot improve the skill through use.
Mythos only improves through encounters with horrors of the Mythos (by way of insane insight) or exposure to the dread literature that describes it.
I think you might be confusing the system with something like Burning Wheel/Mouse Guard in respect of improvement through failure - or perhaps simply misremembering. CoC only supports improvement where the character succeeds.
Does Cthulhu Mythos get subtracted from your sanity as it did in previous editions?
Yes. A character’s Sanity may never be higher than 99 minus his or her Cthulhu Mythos skill. The more you know of the horrors outside of our dimension/perception/time, the weaker your grasp on sanity becomes.