Caledonian Connections

Doctor Who is a television programme. Television was invented by a Scotsman - John Logie Baird - so you could say Scotland is responsible for everything Who. It would be a bit boring to end the post there, so here are some other Scottish connections in Doctor Who.
The Highlanders sees the Doctor's first on screen visit to Scotland - though we know that he has been there before. In The Moonbase he states that he studied medicine under Joseph Lister at Glasgow University in 1888 - gaining a doctorate. This goes against other comments he has made - saying he is not a doctor of medicine. Lister left Glasgow in 1877, so he does seem a bit confused on the matter.
Had Henry Lincoln and Mervyn Haisman not fallen out with the production team over The Dominators, there was to have been a third Yeti tale set in 18th Century Scotland, where Jamie would have returned to his ancestral home and become Laird - leading to him leaving the TARDIS.

The second televised appearance of Scotland appears in Terror of the Zygons - set close to Loch Ness. The famous monster "Nessie" proves to be a cyborg Skarasen. To mark the visit, the Doctor dons a tam o'shanter and tartan scarf. In Underworld, he implies that he has been to the 'Granite City' - Aberdeen.

In Logopolis, the Doctor mentions that they still have Police Boxes in the north of the country. By this I am assuming the north of Great Britain - i.e. Scotland - as there are many still existent in Glasgow and other places. The one above is in Buchanan Street, Glasgow. There are several websites devoted to Glasgow Police Boxes. As the telephone was invented by Alexander Graham Bell, we can thank a Scotsman for the iconic TARDIS appearance - in a roundabout way.

The third trip to Scotland is a brief one - when the Sixth Doctor pursues Vena there to retrieve Tekker's power key. Once again he is close to Loch Ness. He meets H. G. Wells, and the young writer stows away onboard the TARDIS, ending up on Karfel where he'll be inspired to write The Time Machine. Timelash tries to imply that the Loch Ness Monster might equally be the Borad - but I like to think the Skarasen probably ate him.

The last on screen visit to Scotland (to date) is in Tooth and Claw. The TARDIS arrives on a remote moor and the Doctor and Rose meet Queen Victoria. The Doctor pretends to be a Scottish physician - Dr. Jamie McCrimmon, from the township of Balamory. The Torchwood Institute owes its foundation to events in this story - named after Torchwood House. We later learn that there is a Glasgow branch of Torchwood run by a strange individual. He must be very odd indeed to be thought "strange" by Captain Jack...

Companion wise, there is a very strong possibility that Susan may have settled in Scotland after she left the TARDIS - marrying a young Scotsman (David Campbell) who is keen to get back to working the land.
Jamie McCrimmon is Scottish through and through, and the Brigadier is very proud of his Scottish ancestry. When first seen in The Web of Fear he is apparently a member of a Scottish army regiment (he's wearing a Glengarry), and later sports a Clan Stewart kilt in Terror of the Zygons.
Though brought up in Leadworth, near Gloucester, Amy Pond is another Scot - making River Song half Scottish as well as part Time Lord.

Even the Daleks know about Scotland. They chose to programme their android Professor Bracewell as a native of  Paisley, Renfrewshire. Of course, Paisley just happens to be current show-runner Steven Moffat's birthplace.
In The Beast Below, we learn that Scotland went off on its own after the evacuation of Earth.
Amy Pond may have now left the TARDIS, but I doubt it will be the last time we see or hear about Scotland in Doctor Who.