Psi attacks depend on a lot of variables and are chance-based, so nothing is ever certain.
But for a mid-tier Ethereal at 20 tiles trying to panic your soldier, the abovementioned 69 psi strenght (and presumably not much psi skill) will land you at
very roughly 40% chance to get panicked. Every extra point of psi strength, 5 points of psi skill or a tile of extra distance gives you roughly a 2% better chance. So I'd say 60-70 psi strength is where it's starting to become viable to resist Ethereals, but even a full 100 in both psi strength and skill won't make you fully safe from a closeby (a few tiles) Superhuman Ethereal Commander trying to get your troops to curl up into small X-balls.
Obviously, Sectoids are a bit weaker and Mind Control is easier to resist.
Bravery will help a lot when tanking psi attacks - you lose 110-BRA morale.
I suspect the toughened-up Cyberdisks are a consequence of the difficulty bug - old X-Com reverted to Beginner when a save was loaded, and thus everything had its armour and firing accuracy halved.
Even so, you should be able to damage Cyberdisks somewhat even with Rifles: a rifle does 0-60 damage, *0.8 for Cyberdisk resistance against AP, -34 for armour. So 0 on average, but at least some damage ~1/4 of the time, ~7 points on average. Cyberdisks have 120 health, so it should take ~70 shots on average.
Probably less, since the armour will get dinged by the occasional lucky shot.
Big terror units stuck in Terror ships were always a thing as far as I recall. You should be able to play with the doors a little. The disks might not be watching the doors all the time, either.
If all else fails, Blaster them or blow up the Terror ship some.
