Do you not want to be tagged by Lucifer at all?: no this is neato
Can Lucifer sense anything supernatural/otherly about your character, and if so, what?: A literal ton of things. Sophie is a soulless skinchanger that is also tainted by the magical essence of Hell. Reading her stats/perms is easier than me trying to explain it, but in her canon she was chased by the Devil every night for five years before fighting him an making him run.
Are there any limitations on Lucifer injuring your character?: Sophie is very hard to injure or kill, due to her nature.
Lucifer is capable of manipulating objects through telekinesis and reality warping. Can he do so around your character? Are they able to prevent this ability?: sure, She wouldn't find it that weird. Sorcery, man.
Lucifer is capable of mind-reading and may want to do so to mess with your character's head. Are they resistant to mind-reading? Are there any limitations to what he can see?: He can read Sophie's mind, but if she figures out he's doing it she'll lash out at him with the same power she hurt her world's Devil with, and it will probably be unpleasant for everyone involved.