When sun rays shoot through my house from a window, sometimes if the lighting is correct, it will reveal a huge amount of floating particles and dust.
Are these particles dangerous to humans if you live with them constantly?
You asked the single most difficult questions ever, so be proud of yourself! I've spent some time in the lab with a post doctorate trying to answer that particular question. It all is based on something called the hygiene hypothesis: the belief that as we are getting cleaner we are seeing less and less bacteria and so our immune system isn't working as it should so we get allergies. There's lot to this theory with regards to the different white blood cells in our body (specifically Th1, Th2 if you've ever heard of them). Of course there are reverse hypotheses, because we avoid eating stuff like peanuts, but minute amounts is found in dust, dust exposure (which also contains bacteria) makes us think peanuts are harmful so that's why we have allergies.
Realistically a bit of dust to the most of us isn't harmful but if it contains things like dust mite poop which many people are allergic to, cat skin (another seriously common allergy) or is leaking from a place with mould... you should be okay :)
Our basic defenses such as nasal hairs and mucus help in trapping the dust particles and other Suspended Particulate Matters (such as smoke etc).
However, if there is too much of dust inhalation then, as you suspect, it will be harmful. This can be seen in quarry workers who inhale a lot of dust- which causes silicosis.
Dust also instigates asthma.
You would feel much more comfortable in a dust free greenery but the mild dust which you would notice in your house wont be very hazardous in usual circumstances.