Well let's break this down;
1) You pour the stuff on you when you have an open cut - ever think about why? The prime reason is to kill bacteria. Most things that kill bacteria also kill us. Booze is no exception there, actually - we just have a decent tolerance to that particular poison, compared to some other ones, but get enough drinks in you, and you can OD on alcohol - at a level about 6 times higher than being legally 'drunk' (0.5% blood alcohol or so).
If stuff kills one kind of life form, it's likely to also be dangerous or deadly to others.
2) Free radicals react with whatever is available, since it's the single most unstable arrangement of molecules there is. Incidentally, it's the free radicals in tobacco smoke which are though to be the cause of much of the cancers - we call it all "tar" but it's a mixture of lots of benzene rings and lots of free radical chains. The reason this is dangerous is because it leads to a chain reaction.
A radical, once formed, can only be "captured" in one of two ways - reaction with another radical (they mutually stabilize each other), or reaction with a compound which forms a relatively stable radical (transition metals such as iron are very good at this; electrons in a sheet of metal are so spread out that it makes almost no difference if the total number of them is even or odd)
The more common fate of a radical is for it to attack the first molecule it bumps into, and steal an electron away. The victim is now a radical itself, and it goes on to do the same thing to the next molecule it bumps into, et cetera. When I say "steal an electron", I mean that literally. If it takes a "free" electron, it creates a radical version of the molecule it attacks - but it can also steal a bonding electron - if it's from a single-bond, the attacked molecule is broken into two molecules; if it's from a double-bond (or higher), you've changed the bond order. Sometimes, the radical forcibly bonds itself to its victim, causing a different piece to be ejected. In any of these cases, you have changed the nature of the molecules, perhaps unimportantly, perhaps drastically.
The "trouble = bubble" reaction is mainly one of the peroxide attacking anything that's not water. Ultimately, O2 and/or H2 gasses may be produced by the multitude of reactions going on, but that's because H2O2 reacts with almost everything, and does so in so many different ways - radical chemistry, acid-base chemistry, straight-up oxidation, straight-up substitution. The concern isn't the bubbles - the gasses aren't dangerous in the quantities produced - but in what's leftover - lots of OH groups attached where they shouldn't be, and possible scrambling of other parts.
The concern about what's left over depends entirely on what you started with. When you pour it on your skin, well, you kill off a bunch of bacteria, you kill off some of your own skin cells (but as they were already damaged, that's no great loss), and your body knows what to do with dead bacteria and dead skin cells - digest em and make new skin. In the 5% concentration we use, it's not deadly on the skin - especially since skin is a very poor absorber of chemicals (its biological purpose, in fact, is to keep outside chemicals outside, and it's very good at that). The lungs, on the other hand, aren't quite as good at keeping out corrosive chemicals - they're meant for filtration of air, not being a physical barrier. So when you breathe in something that's bad, it's more likely to cause harm, than when you merely touch something that's bad. Incomplete H2O2 reactions, and whatever soup the H2O2 made, would be why peroxide would be a good choice to disinfect, but a bad choice to inhale.
3) That leads to the last one - why does hydrogen peroxide go bad. There are a couple degradation pathways I can think of that would cause this. The most likely is the self-reaction side chains.
Whenever a chemist says something like "X plus Y makes Z" they're not necessarily lying, but they're usually not telling the whole story either. The X+Y->Z reaction is probably the most important, or the strongest (sometimes those are not the same thing), but it's not the only possibility. The rarer possibilities are left unmentioned for short time scales, because, well, they're rare. But over a long period of time, their effects can add up.
I suspect there's a relatively rare H2O2 + H2O2 -> O2 + H2O + H2O type reaction going on. I say it's relatively rare, because it's got to be a multi-step reaction, and your typical bottle of Hydrogen Peroxide is, what, 5% peroxide (and 95% water), so the odds of two peroxides hitting each other is about 1 in 20. The odds of them doing it the right way are higher. But if one peroxide steals a proton from another peroxide, it should release a water molecule shortly after, leaving an unstable HOO group, and an OH group - and the problem there is, one of the two has to have a formal positive charge (Oxygen almost always carries a negative charge, when charged), so this would be a disfavored reaction pathway If the OOH holds the positive charge, however, you'd release O2, and form an H3O+ ion, which IS stable in water (it's the main unit of acidity).
I suspect that pathway is the most likely fate of old peroxide bottles.
In summary, though, I'd use a layman's analogy here; alcohol is a butler who knows how to kill. It's good at cleaning up stuff, both living and non-living - but in highly pure levels, it cleans "too well" and kills - which is good for what we use it for (other than its recreational use, naturally). Peroxide is more like a hitman - it's absolutely lethal to bacteria, but it's not all that great at cleaning up - no better than water, really, at cleaning. (That said, water itself is an excellent solvent, otherwise life could not exist - but when we "clean" we're usually dealing with stuff that water is bad at handling, because water has already dealt with the stuff it's good at)
