"Smells like something died" - well, something DID!

Fourteen days ago, our pet hamster escaped. We could hear him but not reach him for the first day, then silence. Okay, so he died. Sad, especially for the kids, but no big deal really. We thought.
Then, four days ago - ten days after the disappearance, I noticed a faint bad smell in a second-floor room, directly above the family room where the hamster cage was (and, in several past escapes, he'd always confined himself to the family room). It smelled a bit like garbage.
Bit by bit, the smell has overtaken the house. It's still faint, but still garbage-like, and it is strongest in the rooms at the back of our house, which is the side of the house from which the hamster was last seen.
We started to put two and two together, and just for grins, checked my furnace filter today (wondering if I'd see any hamster hair or whatever).
The filter was replaced six months ago and looked like it had been there for 20 years. This is a (roughly) six inches thick by three feet wide by three feet long hepa filter, accordion-folded hepa paper. There was pink insulation all through the filter, bulging the corrugations way out of shape. The top edges of several folds were chewed through in a rounded way, like rodent chews.
We've got our HVAC guy coming this week to clean the ducts in the house, which have never been cleaned, ever. I'm hoping that this gets a lot - dust, dirt, and especially the hamster.
1. What are the chances that the mess in my filter came from the hamster, vs. some random rodent intruder?
2. How the heck could whichever culprit have gotten into my furnace filter?
3. What are the chances that the duct cleaning will pull the deceased rodent from the system?
a) We have a heat pump heat system, and when I say "furnace" I'm talking about the inside basement unit.
b) So the temperature in the duct wouldn't have been hot enough to fast-kill the rodent.
c) There are floor vents in the house, maybe 12 inches by two inches, and wall cold-air returns. The fins in these things are (guessing) 3/8" apart, and I can't imagine our hamster fitting into them.
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