When Mobius steals Renslayer's device he puts it in his inside jacket pocket but when he's alone he takes it out of his trouser pocket.
Loki says, "I've lost track of the number of times I've been killed, so go ahead, do your worst." Assuming that he had not been killed numerous times and then resurrected repeatedly prior to the events of Thor (2011), then his statement is incorrect as this version of Loki was brought to the TVA well before his many deaths and returns throughout the span of the Marvel films.
When B-15 takes Sylvie to question her about also being a variant, Sylvie uses her power, which would be impossible because she has the collar on which negates her abilities.
This is not what the collar does, its a time-shift device. Mobius kept putting Loki back in his chair when he was questioning him, remember?
Something about the area that the TVA is in prevents their powers from working. It is why B-15 had to take Sylvie out of the TVA facility for her to use her magic.