The appearance of new genes can be tracked by sequencing of populations and the appearance of the gene in the population. It involves looking at the specific genes and homology to other conserved genes in a population. Often it also involves tracing migration and behavior of humans in the past. Then you often see a correlation with human migration and settlement in a environment where such a mutation is beneficial.

What do you mean where it originated? A mutation is stationary in a gene and doesnt move. Should a mutation occur in another location it is a seperate mutation. You cannot really time a mutation, however you can determine the homology between genes and their differences (could be referred to as mutations) and somewhat give an estimate when the genes went from one to become two seperate similar genes. If you are taking somatic mutations like those given rise to cancer, they always appear mostly within a decade at the most before the cancer becomes apparent to the individual.

Many oncolytic viruses does target other cells as well however they are not strong enough to infect or spread infection in them as the immune system mechanisms in healthy cells recognizes them and initiates defenses. However, as many cancer cells often turn off immune responses in themself as part of their cancer mechanisms they cannot defend against the virus and is killed as a result. Oncyltic viruses are often insect virus as far as i remember meaning they do not have mechanisms to avoid mammal immune mechanisms. They have showed promises and is a viable means, however as far as i know they are yet to be used in human trials.

Kill HIV-2? Do you mean detect? HIV-2 is a another type of virus than HIV-1, therefore for it to detect both it would need target a common motive for both. HIV-1 test would not be made to detect HIV-2. You can get false positives and negatives in any test, and yes some might lead to more false negatives but the test can still detect them.

I would presume modern HIV PCR detection uses a variety of primers, that would in all likely cases detect the presence of the virus. The same is possible for the antibody test. However, it could also be that both tests target a gene or protein region that is so highly conserved it would never change. Thus any mutation would not be able to avoid detection. Any mutation significant enough that no primer or antibody would not effectively detect, would in all cases also affect the virus to a degree it would most likely be non viable. Even something as mutagenic as a virus or bacteria needs some conservation of specific regions to function properly.

Often it is done by knocking down or knocking out the genes and seeing the effects. So simply seeing what happens if the gene isn't their to produce a protein or RNA species. Sometimes you can also look at what other genes or gene products they interact with and seeing the effects of these interactions.sometimes you can also run predictive algorithms using machine learning using previous known protein and functions to predict a new gene/gene product function.