What can I do to keep my knees warm? It's getting chilly out there and my knees are starting to hurt from my short commute. I tried layered trousers (thick jeans + windstopper on top), but to no avail - my knees still get cold, while my whole legs sweat.

I am thinking of some kind of kneepads. If a product could provide (listed in decreasing order of priority): 
(1)warmth
(2)comfortable to wear all day, under baggy trousers
(3)knee joint support, as this is the joint I (ab)use the most
(4)knee protection during falls
it would be perfect!

I tried two types: volleyball sports soft pads, and medical support pads (from the pharmacy, very cheep). The volleyball are wonderfully warm; however both begin to come loose after maybe half an hour of riding.
You can get explicit cycling knee warmers - I have a pair of these which see a lot of use (and sometimes go under jeans, very comfortably, if I'm out of an evening too). They are great for the morning commutes where it is cold but likely to be warmer by the time of the return journey, but also for the longer rides where you start colder than you'll finish and you need the flexibility to remove bits of kit. You can get fleece lined versions which are even warmer. 

They're not cheap - but there are other cheaper alternatives designed to a similar style. 

They don't offer explicit knee support, but if you need support for your knees when going about your day, then I doubt a cycling Q-and-A site is the best place for your question. 

Similarly, if you think your cycling is causing your knee pain, I'd recommend a visit to your local bike shop and ask them about whether your bike is fit and adjusted properly.