Hi Alison,
Regarding the corner, its probably on the borderline timewise as to whether the render repair has worked (unless its clearly wringing wet!). However, I don't think you've got raging rising damp. You could test the area by painting it with a stain block like zinsser 123, and then overpainting with your topcoat - this will only work if the wall is 'dry'.
Corners are also prime areas for
condensation problems - is this possible? is the room well ventilated? was there any
mould forming? is the area covered by furniture? So before getting a damp proofing firm in I'd wait a little and do a bit more testing.
In the spare room (which I'm presuming is upstairs?), that's definitely not rising damp, but could be a bit of damp caused by the chimney. Is it used? is it capped off? is it ventilated? If its been skimmed, and the plaster has clearly dried out, I wouldn't worry too much about the bleeping damp detector - unless there is no chimney ventilation.
Best