Interestingly enough, some of the words in Scots that differ from standard English seem to be cognates of Danish words. Example:
ken in Scots, kende in Danish quine in Scots, kvinde in Danish bairn in Scots, barn in Danish
Are these words some leftover from the days of the Danish rule over part of Britain?
The simple answer to your question is yes, to some extent. At the time of the Danish rule in north-eastern England, English was not widespread in Scotland. However after 1000 when English began to replace Gaelic and Pictish in Lowland Scotland, the Northern English dialects from the area of the Danelaw were the ones which formed the basis of Scots. However since English is a Germanic language like Danish some cognates are to be expected in any case.
As a matter of interest, when I visited Denmark on business a couple of years ago, I got some amusement out of walking around looking at shops and signs for examples like the ones you give. -- Derek Ross