I would still tell them to sling their hook in the first instance, because it makes sense not to rise to any idle threats.
However do not rely on the misconception about "rights" as a sole trader, you don’t have any, if somebody can prove that you have benefited by using their name, they have a case (although it doesn’t mean to say that the case has any value) but more so when it comes to Ltd companies as its easy to prove registration etc etc.
However it takes money as I said before and few bother unless there is low risk and you have enough for them to make it worth their while or they feel that you are a serious threat.
I again agree with Fox regarding the franchise issue. This is the bit that really doesn’t make any sense, as nobody is gonna sell you the name unless the company is redundant or they want to sell the entire company (maybe a franchisee trying to flog his investment).
Until somebody writes to you, I would simply ignore
it.