In the meantime the thought of Dominic Raab on charge is frankly terrifying. Would it definitely be him
It isn't clear. As First Secretary of State he is de facto Deputy PM, so he is standing in for Boris at the moment but that doesn't necessarily mean he would take over if Boris died. It would be up to the Conservatives to select a new leader. Given the current situation, I suspect they wouldn't have a full-blown leadership election. My guess is a temporary leader would be chosen by Conservative MPs or possibly just the cabinet with a full election not taking place until the current crisis is over. But that is just a guess.
So was this actually true
He did say in one of the regular press conferences on 3rd March that he was shaking hands with everyone and that may have included some coronavirus patients. I wouldn't describe his comment as boasting. He was emphasising that the guidance at that time was that shaking hands was ok as long as you washed your hands properly, a stance with which the CMO and Chief Scientific Adviser agreed at that time.
He tested positive on March 26th. I presume that was soon after the onset of symptoms. The incubation period for Covid-19 is 1-14 days, most commonly 5 days according to the WHO. So he was probably infected around 21st March and no earlier than 12th March, give or take a day or two. On that basis it seems unlikely that he picked it up by shaking hands with infected patients.