You could look for the St Cuthbert Poorhouse records which should give you info on the family
.... but I fear you might have trouble finding them. If I recall correctly, the surviving St Cuthbert's parochial board records are very incomplete. Generally the minute books survive, but not the registers of applications or the registers of poor, which are the ones that contain all the interesting personal details.
To see what is available go to
http://www.scan.org.uk/catalogue/ and search for St Cuthbert's Parochial Board. The surviving records are held by Edinburgh City Archives
http://www.edinburgh.gov.uk/cityarchives As for what was going on in Latheron around 1800, you could do worse than take a look at the Statistical Account written in the 1790s. Go to
http://edina.ac.uk/stat-acc-scot/ - click on 'Browse scanned pages' to view it free of charge.
As George Gunn was born in 1857, after the start of civil registration in 1855, there should be a death certificate for his father Alexander, assuming that he died in Scotland. This should tell you the names of Alexander's parents. You can get this from
www.scotlandspeople.gov.uk at modest cost.
I can see why someone might have assumed that Alexander was the son of Robert Gunn and Ann Campbell. Imagine that this person has found a census record, perhaps the 1881 saying that he was aged 69, or the 1851 saying he was 39, and that he was born in Dalkeith. If that someone then looked for a baptism of an Alexander Gunn, in 1812 in Dalkeith, the nearest one to come up in the search is the son of Robert G and Ann Campbell, born in Edinburgh in 1811. They might then have assumed that because this one is the only one in the index, he must be the right one. This is very common, but unfortunately it is never safe to assume. It has been estimated that something like a quarter of all baptisms were never recorded.
There are two children of George Gunn and Elizabeth/Betty Gunn in the index at FamilySearch: Kathrine, baptised 1810 in Corstorphine, and Elizabeth, baptised 1820 in Edinburgh. You can view their baptism records at Scotland's People.