[1] I avoid the BBC comments like the plague, for the same reason I avoid the Daily Mail.
[2] I was born and grew up in England; I've lived in Scotland for the past 22 years.
The Labour party have traditionally been very strong in Scotland - the Labour party was a party for life, and many people only every voted Labour. However, this has all changed over the past year, as they have effectively self-imploded to the point of uselessness. I cannot overstate how much of a breakdown there has been within the Labour party, and people who had previously only ever voted Labour are now voting differently.
I'd argue that much of the Conservative party's gains have come from people who will simply no longer vote for Labour, rather than a move towards the Conservatives themselves.
I've not heard much, if anything positive about May, but Davidson is generally highly regarded, even if very few actually vote for the Tories in Scotland.