What I suspect you're getting at is, rather, if our DNA contains virus DNA, then presumably that virus' DNA also contains the DNA of organisms it encountered (using the same mechanism), etc. Thus, there ought to be an unbroken history -- albeit probably highly corrupted from successive adaptive mutations -- in our DNA to the first living thing. This wouldn't necessarily be everything, though; just everything we've assimilated.

That doesn't mean that everything has already been "invented" at the root.

Exactly. The "classical" view of evolutionary genetics is that new genes develop by mutation, are favored (or eliminated) by natural selection, and exchanged only within the same species by interbreeding. So the family tree of life is a map of how closely organisms are related, and they would share a gene only with the nearby branches of the tree descended since that gene arose.

We have long known that the simple model is not complete, prokaryotes (bacteria and archaea) exchange lots of genetic material outside their own "species" (trickier to define for bacteria in any case), but it is becoming more widely recognized that lateral gene exchange happens a lot in all organisms.

A recent popular book on the subject is "The Tangled Tree: A Radical New History of Life" by David Quammen.

Imagine in languages: Indoeuropean languages share a common ancestor, but we can say that suchandsuch word is originally from some language and then adapted into another language and used for potentially different meaning.

I don't think that's an established fact at all. DNA is a very complex system so it's reasonable to think that simpler systems predated it. Not that it affects your reasoning at all, since every current living thing that we know of does use DNA.