Are there viruses that affect cells across different species?
Firstly, it's important to recognize that "plant viruses" do not exist. There are only "viruses that affect particular plant cells", or "viruses that affect a particular cell type". You'll see why in a moment. 

One of the structural components of many virus is its protein coat. Different types of biological molecules protrude from the surface of this protein coat, deemed signalling molecules. The structure of these molecules are specific to a certain type of virus.

Analogously, specific cell types have specific biological molecules that protrude from their plasma membrane, deemed receptor molecules. For a virus to affect a particular type of cell, their signalling and receptor molecules must "fit within each other", like lock and key. Once they do so, a virus may interact with this molecule, in whichever way it does.

Although your hypothesis wasn't entirely correct, as viral contact with a cell must first be established, viruses do interact with DNA replication in different ways, as in the lytic or lysogenic cycle.