I did all this a couple of years ago but my memory is not great so I'll just answer what I'm sure of. Yes, to Lightroom Classic. I was backing up my catalog to the external hard drive that also has my images. I used my last backup when the new Lightroom Classic asked (I think) for the catalog. I wish I could remember exactly how it went but I didn't find it difficult. Yes you can have Lightroom Classic on two computers. Someone more knowledgable will chime in, I'm sure. As you can tell, I am not an expert and I had no problems.
Paulette