When virus is present in the bloodstream, this is a medical condition called viremia. Primary viremia occurs when the virus first enters the bloodstream from the infection site. Secondary viremia occurs when the actively replicating virus caused by the primary viremia spreads further in the bloodstream.

In general, virus is some genetic material packaged in a protein shell. It blurs the line between the living and non-living things. Some biologists argue that virus is alive by themselves, some other argue it is not living at all. Some argue that the virus combined with the cell it infected is the living organism. However, it is clear that a virus does not function outside of a cell, in this case the bloodstream. Being in oxygenated blood therefore are irrelevant to whether a virus can function.

A virus is kind of a weird thing, scientists aren't really sure if they're considered "alive" or not. As far as can be settled, they're simply functional until they're taken apart or deactivated (and even then, we're still trying to come to a consensus on whether they qualify as "living"). That being said, we can still deactivate viruses and stop them from damaging and infecting host cells.

Going back to the original question: a virus can still work and infect cells in oxygenated blood. Different viruses have their own preferences for oxygen levels. For example, some prefer living in low oxygen levels because their host cells are tumor cells and solid tumors are "hypoxic" (has low oxygen levels).[1] The opposite is true as well; there are viruses that prefer being in highly oxygenated blood and are actually slowed down in low oxygen levels.[2] Interestingly, viruses like COVID-19 can actually cause low blood oxygen levels although we are not exactly certain why.[3]

As an aside, scientists are developing a method of weaponizing oxygen against infections and disease. By using small compounds called "photosensitizers", you can convert oxygen in a organism to a reactive oxygen species like hydrogen peroxide. These species then react and destroy infectious bacteria by destroying the cell walls.[4]

[1] https://www.microbiologyresearch.org/docserver/fulltext/jgv/96/8/1979_vir000172.pdf?expires=1631726847&id=id&accname=guest&checksum=552DDB06408A2FDA94856890017733BA
[2] https://virologyj.biomedcentral.com/articles/10.1186/s12985-020-01374-2
[3] https://www.bu.edu/articles/2020/3-reasons-why-covid-19-can-cause-silent-hypoxia/
[4] https://www.acs.org/content/acs/en/pressroom/newsreleases/2018/august/weaponizing-oxygen-to-kill-infections-and-disease.html

Yes - viruses that infect oxygenated blood can persist in oxygenated blood. Viruses don't exactly 'live' - there's even a debate about whether they are 'alive' or not. I think that 'living' is a broad category, ranging from things that are clearly alive, down to viruses, which can only reproduce when they're in a living cell. Then there are also seeds - if you try to germinate the seeds, some will sprout and grow, so they were alive. Others won't sprout or grow, so they have died at some point. And there are parasites that can only grow when they're inside some living organism.

I think viruses are sort of like that, except that viruses are even more 'helpless' than other parasites, because they need to use enzymes in the cells they invade in order to reproduce themselves.

I have a friend who co-authored a book about Astrobiology. He agrees with me that viruses are alive, but the other author of the book thinks viruses are not alive. It all depends on how you define life, and there are many scientific papers written about this question.

Viruses in blood would need to get into a cell in order to reproduce themselves. Hepatitis viruses can be found in the blood. They reproduce themselves in liver cells. That's why hepatitis is a liver disease. Hepatitis can make your skin and the whites of your eyes turn yellow, because the liver isn't working well enough to remove the yellow pigment from the body. I know a man who felt sick and turned yellow - he thought he turned yellow because he was drinking too much carrot juice, but that wasn't the reason for his yellow color - he had hepatitis!

Here's a scientific paper that answers your question:

"Generally, viruses that naturally infect well-oxygenated organs are less able to infect cells under hypoxic conditions. Conversely, viruses that infect organs under lower oxygen tensions thrive under hypoxic conditions. Jul 27, 2020"

ODD: Oxygen dependent degradation
RCC: Renal carcinoma cells

A specific virus might persist in blood for a given period of time, but in order to replicate, it must enter a host cell. The level of oxygen in the blood potentially could influence how readily a virus can enter a host cell or how effective a host immune cell may be at recognizing and clearing the virus. The details of this will be largely dependent on what type of virus it is. Not every virus can survive every type of condition (thankfully!).

Different viruses have the ability to persist for different lengths of time in various conditions; the key thing is that they be able to enter a host cell, so if that doesn't happen within a given span of time (again, different for each type of virus and under various environmental conditions), the virus will "die."

Yes. This is partially because the blood plasma itself is not actually oxygenated - the oxygen in your blood is bound up to the hemoglobin molecules in your red blood cells, and not simply dissolved within the blood plasma. For this reason, viruses in your plasma are not exposed to the oxygen in your red blood cells.

Also, keep in mind that some viruses are capable of surviving in air, which also contains oxygen.