Is the nuclear envelope present in G1 of interphase? Is the nuclear envelope present in G1 of interphase of eukaryotic cell? If so how does a method like Calcium mediated transfection get DNA past the nuclear envelope?

To answer the question about nuclear envelope in G1 phase, yes it exists. The nuclear envelope is reformed at the end of Telophase after the 2 copies of the DNA are pulled apart, but before the two daughter cells are separated by cytokinesis. We can see the new nuclear envelope in these images: !reappearance of nucleus in telophase

Image from here

!Nucleus present in cytokinesis

Nuclei visible near end of cytokinesis, centrosome still visible. Image from here.

As for how DNA gets into the nucleus without cellular division I don't have a good answer and I'm not sure anyone does right now. The nuclear envelope is a major barrier for non-viral gene delivery.