Can someone explain DOO in Scotland to me, I've never quite grasped what appears to be an inconsistency in how DOO is defined (particularly in the media)?
There is no inconsistency in the definition; it simply means driver only operation and purely relates to the operation of the train.
I've seen DOO referring to driver opening and closing the doors, or just opening the doors with a conductor closing them.
DOO trains do not have 'conductors'; they may have a member of staff who is present to check/issue tickets and provide customer assistance but they do not get involved in the operation of the train.
Trains which are not DOO have Guards; that doesn't necessarily make them 'conductors', though in Scotland all Guards do check/issue tickets, though this is not the case on all TOCs.
In railway terms (ignoring trams!), the term 'conductor' is generally used to apply to a commercial Guard (i.e. a Guard who also checks and issues tickets); not all Guards are 'conductors' on some TOCs (e.g. Merseyrail, SWR inner suburban services, etc) while not staff who are rostered to check/issue tickets on trains are Guards (e.g. Lumo, Scotrail Strathclyde electrics, Southern services with OBS etc).
And where there is no conductor there is a ticket issuer/collector (that person may have a different title I have not captured here).
The Strathclyde electric trains are rostered for a travelling ticket inspector (TTI), though if they are unavailable the train won't get cancelled.
Is the conductor also issuing/checking tickets and so one and the same person?
On Scotrail DOO trains the TTI issues/checks tickets.
This is similar to the staff on services operated by Lumo, GTR Southern services with an OBS (On Board Supervisor), Southeastern services with an OBM (On Board Manager) etc; on Scotrail if the TTI is unavailable, the train doesn't have to be cancelled. On some TOCs the unavailability of the equivalent staff member would result in a cancellation (e.g. Southeastern High Speed; the OBM is required in case of an evacution being required in the HS1 tunnels).
Or are there trains operating with a driver, conductor and ticket issuer?
It is absolutely possible for a non-DOO train to operate with a Driver and Guard and also have a ticket inspector/revenue protection officer present. I don't know if Scotrail do this, but certainly other TOCs do.