Okay. I sorted out the problem. It turns out Zapier's Survey Monkey → Google Spreadsheet zap is very sensitive to any change in the survey. I had tweaked labels on a few questions and those were the ones that weren't getting transferred (including OpenMRS ID!). Now that's sorted.
Hopefully, the last "problem" will come in a few days when our free trial period of Zapier ends. I'm hoping the transfer zap will still be available (major bummer if it isn't)... but expect we'll lose multistep zaps, meaning I won't be able to notify the grader when a new entry arrives and have to configure the grader to run every 10 minutes or something like that.
In any case, I'm crossing my fingers and hoping the last of the bugs have been squashed.