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.
Hit count (or count of hits), not hits count.