Edinburgh doesn't really have its own unique accent. The local accent is basically part of the Lowland Scottish, which covers a large region, pretty much half of Scotland. (This is in contrast to the Glaswegian accent which is distinct from anything surrounding that city.)
And of course, the people of Edinburgh come from all parts of Scotland, given that it's the capital; and not only that, but they come from elsewhere in the UK, and elsewhere in the world (eg. yourself!). So it's no wonder you hear different accents among the local residents, just like in London you hear a bit of Cockney, quite a lot of Estuary English, some RP, but also all other British accents (not to mention every imaginable foreign one).
Edinburgh has several accents. Find several people, from different social backgrounds, who were born and grew up in Edinburgh, and listen to how they speak.
Edinburgh has many accents, anyone with a Morningside accent in Wester Hailes at night might have a problem.
You will meet people from Glasgow, elsewhere, and England all living in Edinburgh
Yes. Sean Connery has one.
Yes, it does.
YES