I have never come across the "[salutation], [name]," format and it looks very strange to me. Certainly it is not the normal way to start a letter or email. "[salutation] [name]," is more normal. (I do not believe the former is correct, but cannot cite any sources to back me up at present)
Generally, a comma should only be used where you would pause if speaking (or reading the letter aloud). Since you would not normally pause between the salutation and the name, a comma should only be used to artificially introduce a pause there. Thus a comma is not normally used.
The comma at the end of the line is a different matter. It does not follow normal rules of grammar (in any other sentence or paragraph a full stop/period would be used) and is used only by tradition. For what it's worth, this varies in other languages; Spanish, for example, ends the first line of a letter with a colon, as in "[salutaion] [name]:".