Also, is there any possibility of the virus mutating in such a way that what makes it COVID-19 from a public health point of view remains the same but its genetic identity changes enough to no longer be recognized by the existing tests?
That's unlikely. These assays look for several specific regions of the viral genome under high selection pressure (i.e., highly conserved) to make a clinical diagnosis (such as orf1ab, N, and S genes, ref. https://www.thermofisher.com/us/en/home/clinical/clinical-genomics/pathogen-detection-solutions/coronavirus-2019-ncov/genetic-analysis/taqpath-rt-pcr-covid-19-kit.html and https://www.fda.gov/media/136286/download for examples), with the aim to minimize test error from off-target effects (varieties of genomic mutations, ref. https://en.m.wikipedia.org/wiki/Off-target_genome_editing) and misidentification with other respiratory viruses.
I can't speak to the other questions in your question with numerical error rates of tests, except to say that controls, procedures, and better PCR primers can be used to increase sensitivity, to get a true positive (ref. https://en.m.wikipedia.org/wiki/Sensitivity_and_specificity). Sensitivity is a real issue for test kits in Russia, where one must have high viral counts (and likely be very ill) in order for a test to give a positive diagnosis (ref. https://abcnews.go.com/International/russia-reporting-covid-19-cases-cover/story?id=69717763)
According to PCR News, the Russian test is based on the same method used by other countries.
But the Russian test, PCR News wrote, only detects the virus when there are over 100,000 copies of it per milliliter in a sample. That is far more than in other countries’ tests. A test in use in the U.S., for example, will pick up the virus with just 6,250 copies.
“That would mean it’s about 10-16 times less sensitive than what’s available in the U.S.,” Carmen Wiley, president of the American Association of Clinical Chemistry, told ABC News by a phone. At such a level, she said there was a risk the Russians were missing cases, in particular where people were asymptomatic.
From a public health standpoint, I suspect diagnosis of sick people is less useful than diagnosis of asymptomatic carriers who feel fine but are shedding virus everywhere. You want a test to let you quickly tell positive people to self-quarantine, before spreading it around.