Her mother may well still have been alive when a tutor was apponted. Remarriage was common with the high mortality rate. Might be worth looking for traces of a second husband at some point.
There are quite a lot of sources available for Edinburgh genealogy. One set of my own ancestors moved to Banffshire from Edinburgh in the 1570s. I have been able to trace them (Adamsons and Turings) back to the early 15th century in Edinburgh with relative ease. They were certainly merchants and were often bailies and councilors and had dealings with the crown, but sources where I found them include the Register of Sasines, the Register of Deeds, the Register of the Great Seal, the Register of the Privy Seal, the Acts of the Lords in Council, the Acts of Parliament, the Lord High Treasurer's accounts, the Exchequer Rolls, the protocol books of the various notaries, the Register of Charters of St Giles, the Register of Charters of Holyrood Abbey, the Burgh accounts, the Burgh Council minutes. Many of these have either been published in full, as extracts or have indices published. I have found that if you look everywhere it is surprising what you can find.
If the late George Gawy, tailor in the Canongate, who was dead by October 1607 with young sons, William and John, was still alive in April 1606 and if he was a relative of your Robert, then he is obviously not as close a relative as William in Scrogtounhead. That's two ifs, of course.