I thought the remote possibility that the blood in the silencer could be a mixture of Nevill and June's came post-trial, but I may be wrong. As for the bicycle I have no idea whether Jeremy borrowed or stole it.
Nor I, Steve, so I've had to weigh up the probabilities of both, whilst considering how likely it was that JB and June weren't speaking at all. If they weren't -and for me, it's a very big "IF"- then it's quite likely that JB might have stolen it out of spite. However, from what we know of June, I find it more than hard to believe she'd have cut ties with her son, whatever he'd done, to the point at which conversation was zero. I think she'd have been sad and disappointed, but maintaining total silence, on a daily basis, especially when JB visited WHF regularly, if not daily. I don't think such behaviour can be applied to June. Of course, I can't vouch for JB, but there would certainly have needed to be a conversation had he asked to borrow the bike for Julie.