You could always make up an extract solution, so that you'd know exactly what the gravity should be, and use that as a cross-check of your measurement process.
As Sherlock Holmes said; once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.
The hydrometer is touching the bottom of the flask.
I've been wrestling with the ultra barrier for a week now. On the very same keg the 10ft vinyl line I'm replacing is pouring fine but 15ft of ultra barrier is all foam. I've seen other people having problems that stabilized over time, but I haven't seen it yet.
Even at 154F I'd expect to get significantly lower than 1.022 (and routinely do). My suspicion would be the grain bill not containing enough diastatic power.
A second-hand keggle would likely set you back $150-$200 (depending on the added hardware & welded vs weldless couplings), and a 15G mash/boil kettle and 8G sparge kettle to dunk the bag in after mashing gets you comfortably up to 10G batches.
I'd also go with the dunk sparge - it's what I do with all my 10G batches in 15G keggles. With an 8G kettle to sparge in, you'll probably need something like a 10G mash/5G sparge split of the ~15G water you'll need.
Again, while this will affect the initial hoppiness the OP says that the beer is a hop-bomb to begin with and that the issue is the rate at which that hoppiness subsequently decays. I don't see how the dry-hop timing can affect that; oxygenation is the only thing I've seen suggested that can...
That's exactly what the graph shows!
Since it's a log-linear graph (with the y-axis showing powers of 10), exponential decay is a straight line. Follow the yellow (15 psi) line, and after 10 cycles the remaining 02 has dropped by a factor of 1000 ~ 2^10.
No - they can be used, but you have to correct for the alcohol in the sample so you need to know the initial gravity too.
Plenty of calculators out there for this, for example:
http://www.brewersfriend.com/refractometer-calculator/
You can certainly use an airlock, but there's no guarantee that what you're sucking in has been sterilized in any way. If you're not concerned about infection you can also just cover their kettle and drain it the next day.
Ideally you want a container that holds a seal (to keep airborne nasties out) but can deal with the volume reduction as the wort cools. HDPE deforms, metal buckles.