There's a text for Poor Roger in an earlier discussion:  Poor Roger. That particular one is an example from Ireland, but it was common throughout Britain and the North of America in various forms and under numerous names (number 797 in the Roud Folk Song Index).

The second one turns up as Jenny Jones, Jilly Joe, Janet Jo, and so on. Again, it was quite widespread over the same sort of areas (Roud 1047). There's another children's song, Jenny Jenkins, which retains the basic form while dropping the plot, so there is no death or funeral. I don't know if it derives from the other, but it wouldn't be surprising. Several examples are in this thread:  Jenny Jenkins.