Was it a constitutional monarchy? How was it run with their king in London? Did they have a parliament similar to the English parliament? How many houses was this parliament made up of? What were the names of these houses?
The Scottish parliament was quite different to that of England.
The largest difference was that there was only a single house, unlike England with the House of Commons and the House of Lords.
It wasn't the most simple unicameral system as there were three groups known as the "Three Estates" sitting in one chamber. Initially the three estates were made up of
1st - the clergy
2nd - the nobility
3rd - representatives of each Royal Burgh
In the 17th century there were several reforms to the Estates, most importantly the 1st Estate was abolished and the 2nd Estate being reformed and offering positions for lower nobility in the form of "Shire Commissioners".
These reforms meant that at the time of the Union the Scots had a single house made up of several separately appointed groups.
It's perfectly fine if you want to delete my comment as soon as someone more knowledgeable comes up with sources. Until then, perhaps my response can provide some information to you.
I'll just point you to http://en.wikipedia.org/wiki/Kingdom_of_Scotland
I'll preface by saying that my comments below really apply only so far back before the acts of union. The Kingdom of Scotland existed for an extremely long time and would've had very different forms of government over that time.
It was run as a separate kingdom until the acts of union occurred. Think of how Canada is run as a separate country to Britain, despite the monarch living in London. That being said, at the time the monarch exercised a lot more power and wasn't merely a figurehead. Despite this, the King's Scottish forces ended up fighting the King's English forces quite a bit, among other interesting situations.
As for your questions: Constitutional monarchy didn't exactly exist at the time, it's hard to explain and I'll leave that to someone else. It really would've been a matter of what the monarch could feasibly accomplish without the gentry in parliament giving him their support, as the monarch didn't really have the physical capability of raising tax revenue by himself.
The king being in London didn't really matter to be honest, from what I understand.