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Hi, I'm Mike G and I'll be glad to assist you. Unfortunately you won't be able to originate a new light from that opening. There is no neutral available. The white wire was used for a hot wire.
Yes you can but you need a neutral for the light to work.
The white wire would terminate on the black screws and the existing black wire and a new black wire on the gold screws of a switch over switch. But you still need a neutral.
That would be best. You're welcome.