Not really, the tests only test for specific proteins found in the virus, likewise the vaccines are designed to produce those same proteins to trigger an immune response that will work against the virus. It's therefore fully expected that a test designed to detect certain proteins would detect those same proteins shortly after vaccination.

That's not quite how it works. Ordinarily what happens in your cells is that your DNA produces strands of RNA, the RNA then acts as a template to construct specific proteins that you need to live. RNA has a limited lifespan and degrades naturally in your cells, but ordinarily your DNA keeps reproducing more RNA to make more proteins.

Now in the vaccine, they inject strands of RNA that your cells don't produce naturally, this RNA acts as a template to produce the proteins found on the surface of the virus, these proteins are then constructed by your cell and trigger an immune response that will also work against the virus. Because your DNA doesn't know how to produce that RNA, it doesnt produce any more of it, and the vaccine RNA degrades naturally away after a couple of days and your cell will stop producing that protein.

At no point in any part of this process were your genes or DNA altered nor were any viruses produced.