Ramshackle Power Armor is stolen from Roz on Zheng Shi (and disables her encounters). I don't think it's worth using because the -10 Evasion basically locks you into using the Light Jetpack or the Rouser (-total Evasion = +0.03% damage/point). Don't even think about using 'Power Armor' weapons with actual Power Armor, because the penalties stack. Technically, there's also the Akkadi Armored Depthsuit from a mission Dr. Lessau gives you.I don't believe I've ever encountered the Ramshackle Power Armor! I also don't believe I've ever managed to trigger Kiro Quest, Kashima Incident, or the Hnngularity. Maybe that last one. It sounds vaguely familiar.
The Distress Calls happen if you're travelling between planets. Kiro Quest requires her Trust to be 100% and level 9+, Kashima requires you to be travelling to or from Myrellion around Halloween and level 7+. I think there's also a tiny chance outside that. 1%?