Reverse transcriptase is a viral enzyme. Normally you cant go RNA-> DNA. Retroviruses use it, there are only two retroviruses known to infect humans: HIV and HTLV. Both of those viruses manage to keep their genes integrated in at least some cells forever, which is why curing HIV is very difficult despite effective treatments that prevent viral replication.

However other viruses that cause persitent infection do not use reverse transcriptase. Some (like HPV) can integrate with the genome but they are DNA viruses integrating with DNA (and not retroviruses). Herpes viruses generally dont integrate with the genome but a few can (again DNA to DNA), but chickenpox (a herpesvirus) as well as herpes simplex (the one you probably think of when you think of herpes) infect nerve cells and hide out in them as the immune system usually does not attack nerves, so they persist through a completely different mechanism from genome integration.

>> How long is the new genetic code part of that DNA? Forever?

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Yes, forever.

We have quite a lot of DNA injected into our cells from viruses in the distant past.

im pretty sure there is an error correction mechanism that removes genetic sequences which seem out of place or mutations. no idea how likely it is to always work, but infected cells are cleaned up through autophagy, programmed apoptosis (this is like murder and suicide), so viral sequences will most likely be removed at some point. chronic viral infections like aids and herpes are forever though. edit: i do not have a degree or anything in this area, so if you want to know more im sure youtube has simple videos explaining these things