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On the first question of removing an 80 a DP and replace with a Quad 30/50, that is fine.
On the second question, not sure what you are asking there? On the 30 amp feed, you cannot go to receptacles, standard duplex receptacles are only rated for 15 or 20 a breakers.
Not sure what you need to power at the shed?
That determines what you need.
Sounds like a working shop area, seems you need a bit more than 20 amp feeder to me, but I dont know all your equipment and how often 2 pieces will operate at the same time?
If you have a panel, I would use it myself with as large a feed as the budget allows.
You could run a 30 amp 240 circuit to the panel .
Or a 50 amp 240 , the quad would allow either
You run a 240 volt feed, then use the single pole breakers at the panel to power receptacles etc.
When adding a panel in a separate structure, the below is the wiring format. 4 wire power and additional ground rods at the new panel.
You could run 30 amp now and later you could upgrade with little work with the conduit in place.
I understood that you had 2" conduit in place?
If you have the conduit, run another wire fro the ground. That is required today for separate structures along with the ground rod. Both are required.
Ok, add a ground wire from the sub to the shed. At least the shed will be up to date.
Probably the sub was added years ago under the old codes.