Using 3rd Edition Rules: For a controlled objective city to be counted for Objective Victory does the city have to be a supplied city as per 6.1-Cities and Supply. I.E. Does Alternate Supply also apply?
Yes it does have to be supplied by 6.1
Alternate Supply does not apply.
Again, for 3rd Edition Rules, 7.6-3rd bullet: Are the North edge hexes excluded as eligible placement hexes for Soviet Units?
YES. This is what the rules state. East and South edges only.
Edit: do note that now you can trace supply not only across lakes and lake hexsides, but also across sea hexsides (rule 6.2).
Seas as a Supply Source
A unit on a coastal hex can use the sea as a Supply Source.
Units cannot trace to a sea—they must be on the coastal hex
(Supply can be traced across a Sea hexside). The following