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Does the crew still count as a heavy support unit when they lose their guns i.e. can only contest objectives not control them?
I'm pretty sure they do but I didn't see anything in the rules. Those Artillery rules are so wonky.

Were they purchased as a troop unit? Do they join a different unit before deployment? Do they have a special rule that makes them scoring?

The answer is usually no to all of those questions - so no, the best they can do is contest.

They count as whatever type you originally purchased them for. Taking casualties doesn't change that they're HQ, Elite, Troops, FA, or HS.

No, artillery crew always stay as whatever FoC they were purchased for, unless they have a special rule that states otherwise.