If I connect to a specific DogeCoin Node, is it possible to determine the transactions that originate at that node? I would like to know if it is possible to determine what transactions are originating at a specific node, assuming that I am connected to only that node.

This post applies to Bitcoin and DogeCoin as well.
No. And if you could, that would be considered a bug.

Think about it. Say I create a transaction. I send it to exactly one node, and only one node. How can that node convince you that it got the transaction from me rather than that it originated it itself? How can that node know whether I originated that transaction or got it from some other node?