What is the proper grammatical way, say, to name a table column: hit count or hits count? I have a hard time choosing.
"Hit count" is the proper way of saying it. The important thing to keep in mind here is the actual subject of the phrase or sentence you're stating. "Hits count" would imply that count is a verb, and you're trying to say that "those hits count," perhaps in a fencing match or similar context. "Hit count" means you're talking about the total count of hits.
It is definitely correct to say 'Hit Count', and wrong to say, 'Hits Count'.
I believe the reason is that you are counting each 'hit', not each 'hits'.
As mentioned above, if you want to refer to the collection as an object itself (the 'hits'), then you could say, 'Count of Hits' since you are now referring to a property of the (collective) group of hits (which is now a single object), rather than counting the number of hits.
A tricky one, that I suspect most (but not all) native English speakers would instinctively know, but would struggle to explain.