[RPG] How strong is the tressym in comparison to other familiar options?

I think these seems like a lot of fun, and I'd probably allow it in my game as well. I've created custom familiars in the past, and most recently I've used this "5e Familiar Handbook" I found. If you're not sure about your custom familiar, compare it to the others on this list. I like how you kept the climb and fly speeds low, but I think I'd personally do something like this: Speed: 20 ft.; Climb: 40 ft.; Fly: 20 ft. With those movement speeds, you'd have a clumsy-lookin' flyer, but an excellent climber, much like what I imagine a flying monkey would move like.

If we're going with a strict reading, then if you're seeing the medusa's eyes from beyond its range, then you -- the remote viewer -- do not need to make a saving throw.

When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature.

Your familiar will need to make a save, of course, if they are within 30ft of the medusa.

If you are within 30ft of the medusa and use your familiar to see, and you see the medusa's eyes, then it depends on the final clause of that rule: if the medusa isn't incapacitated and can see the creature.

Let's imagine you are a resource-depleted familiar-owner hiding from the medusa in an abandoned building. She is only 25ft away from you, but is unaware of your location and cannot see you due to being in total cover from a stack of crates. Then, you look through your familiar's eyes (a spider) to scout the area, but end up looking straight into her eyes.

In the above scenario, you do not make the saving throw, but your familiar does. It would potentially be a traumatizing event though, being in the mind of a creature as it is being petrified.

Otherwise, if you had failed your Stealth check and the medusa actually can see you, and then you scout via your familiar and end up looking into her eyes through the familiar, then yes you will need to make a saving throw.

Strictly speaking, as long as the four conditions are met, you have to make a saving throw.

1. can see the medusa's eyes
2. starts its turn within 30 feet of the medusa
3. the medusa isn't incapacitated
4. the medusa [...] can see the creature

It doesn't matter where your sight comes from as the first condition is not "that can see the medusa's eyes through its own eyes".

I'm reminded of spells such as Misty Step, which allow you to teleport to any unoccupied space you can see within 30ft of you (even if you're seeing through the eyes of your familiar).

As for the case of controlling another creature, that depends on how you are "controlling" them. If it is through Dominate Monster, for example, then only the creature you have controlled makes the save.

But more likely, you're asking about Magic Jar. If you are possessing the body of another creature in this way (your soul leaves your body and steals the body of another living creature), then from the wording of this particular spell, your soul and the possessed creature's body are considered as one creature. You -- the possessor -- must make the saving throw using the stats of the possessed body. If you fail the save, the body you are possessing becomes petrified. If that body also dies, your soul may leave that dead body and go elsewhere.

As for should your original body be petrified along with the possessed body, that depends on if your true body is within 30ft of the medusa. If it is, then you must also make a saving throw with your actual stats.