Peroxide is directly virucidal through oxidation of the lipid envelope and RNA degradation. Whether lymphocytes would be activated in lung tissue in vivo is entirely speculative.
Sure. Again, this is where the mechanism matters. There are plenty of things that kill viruses; you have to be able to deliver it in a useful way that doesn't kill the patient.
But peroxide is not just some random thing like bleach. It's generated internally from superoxide dismutase in the normal process of scavenging radicals and is also an immune signalling mechanism. As mentioned, it is reduced by a reaction catalyzed by peroxidase so the body "knows" how to shunt it around safely and reduce it when necessary. It's perfectly safe in appropriate dilution on scrapes, cuts etc. The key is not to overwhelm the antioxidant defenses. Lung tissue is much more delicate, but it's not ridiculous to suggest that a very low concentration might be worth testing.
As XKCD points out, a handgun kills cancer cells in vitro. https://xkcd.com/1217/