Just wanted to point out where you’ll find the answers in the REG. They are obviously not directly answered in synopsis as Gabe‘s reply, but these are the answers nonetheless:
Paralyze
“While the paralyzed condition remains, the paralyzed character cannot enter battle.”
Paralyze only keeps the character from entering battle, nothing more.
TC Characters
“Territory class character abilities that are active outside of battle are active while the character remains in the territory.”
Territory
“Each player controls an area in the Field of Play referred to as their territory. A player’s territory is where they can put their Artifacts (the Artifact Pile is in territory), characters, Fortresses, and Sites. The player’s Land of Bondage, where they keep Lost Souls they control, is also in the territory.“