I'm happy to see someone already working on 5e Ravenloft - in fact, this has been the very reason for me to register
The Horror/Fear Checks may still need some work as others have mentioned before I found this site, but with the DMG featuring official rulings on those topics I will hold back on thinking too hard about them until in a few more weeks.
Dark Shadows:
I like the general idea, but I think making them a free pick for everyone will lead to an inflation as everyone will pick them. (A hard mechanical benefit offset by a fluffy disadvantage is a steal.) I consider them to be equivalent to an extra language/tool. Not wanting to mess with the existing backgrounds, my way of handling them would be simple:
Anyone
not taking a Dark Shadow may instead opt to pick proficiency with an artist's tool-set, a gaming set or a music instrument.
(This is of minor use, but it comes with no strings and can be put to use without sacrificing inspiration, making it more interesting for those not interested into the fluffy side of their DS. It also ties in nicely with the Gothic genre by making hobby artists and bonvivant gamblers more common among PCs.)
Cat’s Eyes
You were born with bright yellow eyes and slightly ovoid pupils, which catch the light and appear to glow in dim illumination. This rare trait is found only in folks hailing from Vaasi
-speaking nations. It is most common among the Valachani, but the eyes are sometimes found in Nova Vaasans. The cause is unknown but there are many superstitions. Some believe a cat took some of your soul as an infant, others believe you have the blood of a werebeast, and still others believe you are the reincarnated spirit of a feline
Benefit
You can spend your inspiration to speak with felines. You can communicate freely with cats for 1 minute, as is using the spell speak with animals. The cat is under no obligation to help or share information with you.
This one is very, very specific with very narrow application, and no guarantee to get anything good out of it.
Quick-Fix Suggestion: Just add Cat language to the character's languages, permanently.
Cursed
A malicious hag, dark cleric, or vindictive Vistani has cursed you. You might have been cursed by an evil creature for attempting to stop them, or cursed by a good or neutral individual as punishment for misdeeds. The nature of this curse varies dramatically; you might suffer from misfortune, a wasting disease, a weakened sword arm, repulse people of the opposite sex, or any number of lesser or greater afflictions.
Benefit
Having been plagued by misfortune for years, you have learned to to limit the impact chance has on your actions. You can spend your inspiration before making an ability check to roll 3d6 instead of a d20. If you have advantage on the check, you instead roll 5d6 and use the total of the the three highest rolls.
This one would only be worth it for rolls where you have already advantage, or where an average roll would be sufficient. Otherwise it would be smarter to just use the inspiration to gain advantage.
Quick-Fix Suggestion: Make it best 3 of 4d6.
Haunted
The ghost of a loved one or family member follows you around, seldom leaving your side. You cannot see other spirits and no one else can see your ghost, but you can interact with the ghost as if it were a regular person.
Benefit
You can consult with your spirit, learning otherwise unknowable knowledge. Pick one skill: History, Investigation, Perception, or Survival. Once you pick a skill you cannot change it until after you take a long rest. You can spend your inspiration to gain proficiency in that skill for 1 minute. If already proficient, you instead double your proficiency bonus.
This one suffers from the same fate. Only on level 13+ is this one better than to spend the inspiration to gain advantage.
Quick-Fix Suggestion: Leave the proficiency active until the next long rest.
Redheaded
Having been touched by the fey, witchcraft, or other magical forces you possess a minor magical talent. While not all redheads are touched by otherworldly forces - the majority simply have red hair - all those similarly "blessed" possess vibrant crimson hair. The superstitious believe you are the harbinger of ill fortune, or made a pact with infernal forces.
Benefit
You can spend your inspiration to gain the thaumaturgy cantrip for 1 minute. Wisdom is your spellcasting ability for this cantrip.
This one is very weak - trading in an inspiration for brief access to a purely fluffy cantrip isn't really worth it.
Quick-Fix Suggestion: I'd make that a one-off use of any lvl 1 druid spell, or access to a druid cantrip until the next long rest.
Reincarnated
Caught in a choice of death and rebirth, this is not your first life and you have returned from the grave into a new body. You only dimly recall the events of your past life in a dream-like haze but occasionally you experience intense memories, feelings, and urges.
Benefit
Tapping into the knowledge of a past life, you can spend your inspiration to be considered proficient in a skill or tool of your choice for 1 round.
Alternatively, you can spend your inspiration to recall memories from a past life, potentially recalling some details of a person you have never met or place you have never been.
See Haunted above.
Backgrounds:
I like what I've seen so far, but I think some of the new ones could still use some extra jazz and spice

:
Changeling
[...]
Equipment: A set of artisan's tools (one of your choice), an item , a set of common clothes, and a belt pouch containing 15 gp
The changeling background cries for a weird heirloom that potentially ties into an advanture seed. Maybe an extra roll on the trinket table?
Inquisitor
[...]
Equipment: A holy symbol, an herbalism kit, manacles, vestments, a set of common clothes, and a belt pouch containing 10 gp.
Add to that "a letter from a layperson or low-rank clergy about some heresy or similar shenanigans in a nearby location".