Does it's status as a "state church" have any affect the beliefs or practices of the Church of Scotland? Are there any doctrinal distinctives that make it unique (particularly among Presbyterian churches) that can be traced to it's unique relationship with the state?
Regarding belief, the Church of Scotland is entirely independent of the state in matters spiritual (and is a "national church" not a state church). Regarding practices, the call to be a "national church" means a self-understanding not to neglect areas of the country which are less prosperous or populous. The Church of Scotland is struggling to maintain a system of parishes, with ministers serving every parish (including very rural parts of the Scottish Highlands and Islands).
The Church of Scotland has, at times, criticised the policies of the British or Scottish Governments. The General Assembly of the Church of Scotland has, for example, regularly approved "deliverances" calling for nuclear disarmament - contrary to British Government policy.