Thing is Thompson, they can. However, only when in The UK though in theory, although in practice, if your driving licence says Sir on it, other countries tend to accept that.

Irish honorary knights can use title over here whilst, for instance, American ones can't. (Commonwealth aren't honorary, so full knights and dames anyway.) The Irish clause, despite the error in the Wikipedia entry, denotes the fact that Ireland as part of treaties at the time of independence made allowances for living receivers of honours at the time and put the half half status for future ones as a compromise. To be fair, nobody expected at the time that any Irish citizen would wish to receive a Royal honour, but that was a different age...

Names are whatever you wish them to be. You could call yourself Lord Thompson if you wished over here and so long as you don't infer peerage for gain or favour, you are doing no wrong. See the late Screaming Lord Sutch for details.