OK, based on the data from JohhnyBravo, it looks like the answer to Q1 is "YES" for F2Pool but "NO" for AntPool.  Here's a rough summary based on the last 100 days of blocks:



If anyone's interested in how I did this, I'll give you a brief walk through.  I first put the data into three bins:
(1) blocks produced by the Miner after a small block (0 kB - 333 kB),
(2) blocks produced by the Miner after a medium block (334 kb - 666 kB), and
(3) blocks produced by the Miner after a large block (667 kB - 1000 kB).  
I also removed any data points where the Miner found a block while mining upon his own previous block.  

For a "null hypothesis," I assumed that getting an empty block was the outcome of a repeated Bernoulli trial. I used a Bernouli trial with P_empty = 3.5% for F2Pool and P_empty = 6.3% for AntPool.  

I then asked, if the null hypothesis is true, then what are the chances of getting, e.g., in the case of F2Pool, 34 (or more) empty blocks out of 619 "large-block" trials?  We'd only expect 619 x 0.035 = 22 empty blocks!

The sum of a repeated Bernoulli trial has a Binomial distribution, so I integrated the corresponding Binomial distribution between 34 and 619 to determine the chances of getting this many (or more) empty blocks by dumb luck.  As we can see, there's only a 0.4% chance of this happening.  This suggests we reject the null hypothesis in favour of the idea that "Yes, F2Pool is actually more likely to produce an empty block when the previous block was large."

This also means that the effect I modelled on the weekend is real, at least for miners behaving like F2Pool.  I'm curious though, why AntPool's data is so different.