The Scots leid branched off from Middle English tho, well after 1066. There’s plenty of French in the bonny north’s tongue. Scots Gaelic was the language untouched by the Normans, and quite a few words got transferred into Scots when the Gaelic-speaking populations were forcibly migrated into Scots areas.
Sure. My impression was that Scots and Middle English sort of co-evolved and fewer Norman loanwords made it all the way north, though obviously some did.
Scots Gaelic is another tongue entirely, with no direct Anglo-Germanic descent.
To be clear: Modern English and Scots share the same direct ancestor of Middle English. They’re both about the same age, they both diverged from Middle English.
The Romans made it all the way to Aberdeen, the Normans got up north no problem and did pretty well e.g. the De Brus family of Annandale neé Normany.