Is the nuclear envelope present in G1 of interphase?
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:


Image from here



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.