Incidentally, if you're wondering why this kind of stuff only seems to happen in the brain, in broad terms the answer is this: almost every other cell type in your body has the luxury of committing suicide (technical term: apoptosis) and being replaced by other cells of the same type which are growing and proliferating. For example, if a virus infects one of your intestinal epithelial cells, that cell can trigger it's self-destruct signal pathway (or a cytotoxic T-cell can trigger it externally) and initiate programmed cell death, thereby containing the infection (note that apoptosis functions to effectively demolish the entire contents of the cell, not just tear it apart and release whatever is inside).
In contrast neurons generally do not proliferate much after you're born (there's some debate about this, but even so a particular neutron is valuable because of the connections it has made, which are irreplaceable), so having each neutron kill itself when something goes bad would shortly leave you without a functioning nervous system. So neurons have to resort to more "creative" measures to clean up problems without destroying their entire cell bodies (e.g. autophagy, in which the cell cordons off a part of itself and then digests it). Naturally, these mechanisms are not as effective as demolishing the entire cell, so some problems can arise that neurons can't handle. But they are programmed not to kill themselves under almost any circumstances, so they continue limping along with only partial functionality or none at all.
Or, look at things from the opposite direction. Most cells die fairly quickly (in weeks to months) relative to the life expectancy of the organism (decades, for humans). This short life is probably terminated by the buildup of obstructive particles, such as plaques of misfolded proteins, to the point where the energy required to "repair" that cell exceeds the useful functionality that the organism could derive from it by keeping it fixed up. (The analogy would be a car worth $300 requiring a repair that costs $1000.) At that point, it is now "cheaper" to just kill that cell and replace it with a new one. And so this is how most tissues in an organism work: constant turnover and production of new mature cells from tissue-specific stem cell precursors (as opposed to "pluripotent" stem cells which can in theory produce any cell type, not just a certain tissue).
Neurons, in contrast, are special in that they are not replacable, since a "replacement" neuron would have no way of recreating all the synaptic connections made by the old one. A neuron's connections constitute information storage that would be irrevocably lost if that neuron were to die. Hence, neurons have evolved to spend lots and lots more energy keeping themselves alive compared to a typical cell in a multicellular organism, because the information stored in that neuron is (on average) very valuable to the organism. But life didn't evolve with neurons in mind, and it certainly didn't evolve with 100-year-old metabolically-active cells in mind, so this gets harder and harder as the cells get older, because they are pushing the limits of what cells are capable of.
Autophagy is a repurposing of the same cell machinery that is used to engulf and digest large pieces of external material. Instead of extending a membrane outward from the cell surface to engulf an external particle, it forms a membrane inside the cell and uses it to engulf some part of the cell's contents. Either way, the result is a membrane-bound bag of stuff, which is then merged with the cell's lysosome, which is full of acid and digestive enzymes.
Autophagy is a fairly recently-discovered mechanism (compared to, say, apoptosis), but it already seems to be a universally-used pathway, for everything from killing invading bacteria, to clearing plaques and aggregates of misfolded proteins (such as those that cause neurodegenerative diseases), to self-cannibalizing unnecessary cell components for energy and building blocks during starvation conditions.
There's some evidence that autophagy is protective against a wide range of neurodegenerative diseases, so a good bit of research is focused on how to stimulate autophagy in neurons in a safe and controlled fashion (since obviously too much autophagy will kill the cell outright).
I'm not super familiar with the current literature on autophagy. I'm a bioinformatician, and I collaborated on an autophagy-related project a few years ago, so I had to read up on it then. I only really recall the overarching themes at this point. But I do remember that since autophagy is used for so many disparate purposes, and because it could easily kill the cell if it got out of control, the regulatory machinery controlling it is almost certainly quite complex.
However, that paper says nothing about neurons, yet it seems to the basis of most of the "Could excercise stimulate neuroprotective autophagy?" speculation that I can find via Google. So I don't know that there's any published link between excercise and neuronal autophagy. I suspect that it's just the paper above showing a link between excercise and muscular autophagy, combined with the untested hypothesis of "Maybe excercise stimulates autophagy in other tisues too."
By the way, note that while the mentioned study was published in January 2012, it was submitted in September of 2010, which means all the research described in the paper happened before then. I wouldn't be surprised if the current research by that same lab was focusing on establishing a link to autophagy in neurons.
Does any flagging take place in the brain at all? I have tried to keep up to date on Alzheimer's without knowing much about the human body. One thing I've thought about given the debate over whether Aβ is the culprit, one of many culprits, or just a visible sign of Alzheimer's got me thinking about it as a flagging mechanism. If a synapse is dead or in some way misbehaving it might be good to flag it as "corrupted", might that be what Aβ is doing at all? I mean system-wise it makes sense, but I know way too little about the human body or the brain.
There are lots of "damage-sensing" mechanisms at all scales within the body. These are how your skin knows to grow after an injury, how your immune system detects which cells are infected and need to be killed, and how your cells know which parts of themselves to digest via autophagy, among other things. At the single-protein level, there's ubiquination, which marks protein molecules for destruction, either because they are misfolded, damaged, or no longer needed. But this doesn't work when those misfolded proteins clump together into plaques. Then you need something like autophagy. But even that has limits, and a cell that lives for decades tends to reach those limits.
As for Aβ possibly being the body's way of flagging problem areas, that seems unlikely, because such a function probably would not involve drastically changing its 3-D structure (i.e. going from "correctly folded" state to "misfolded" state), and certainly would not involve forming plaques. These and many other observed features are better explained as features of the pathology and not just a kind of damage signal reporting on the existence of a problem.
The linked study demonstrated that injecting amyloid into one part of the mouse brain caused more amyloid to appear in another part. In this sense, the amyloidosis was "transmitted".
The prion-like behavior mentioned is the ability of prions to do something similar. Specifically, a small amount of abnormal prion protein can induce a larger amount to also become abnormal (with respect to it's shape). The process is analogous to crystallization, where small crystal can seed the creation of a larger one.
In the field, the significance of this is that small initial abnormal events can trigger larger ones within the brain, a bit like the famous butterfly analogy in chaos theory.
I believe that Alzheimer's has different clinical presentation than either BSE and CJD, and that the misfolded protein being blamed for Alsheimer's is a different protein than those blamed for the other two diseases.
In contrast neurons generally do not proliferate much after you're born (there's some debate about this, but even so a particular neutron is valuable because of the connections it has made, which are irreplaceable), so having each neutron kill itself when something goes bad would shortly leave you without a functioning nervous system. So neurons have to resort to more "creative" measures to clean up problems without destroying their entire cell bodies (e.g. autophagy, in which the cell cordons off a part of itself and then digests it). Naturally, these mechanisms are not as effective as demolishing the entire cell, so some problems can arise that neurons can't handle. But they are programmed not to kill themselves under almost any circumstances, so they continue limping along with only partial functionality or none at all.