Short answer: Gardens are your best bet — Trebah, Glendurgan, Trelissick and Heligan all welcome dogs on a lead. Zoos and wildlife parks never do. Castles and historic houses usually allow dogs in the grounds but not indoors. Bodmin Jail is the rare indoor exception.
Cornwall is a genuinely good place to bring a dog, but the rules vary a lot between attractions that look similar from the outside. Two National Trust gardens ten miles apart can have different policies. Here's the whole picture in one place, so you can plan a week rather than checking site by site.
There's a logic to it that makes the individual answers easier to predict:
Bodmin Jail is the standout — one of very few indoor Cornish attractions that welcomes dogs everywhere, which is worth knowing on a wet day when everything else is closed to them.
Trebah goes further than most gardens by allowing dogs on its private beach at the foot of the valley. Heligan and Trelissick both cover enough ground for a proper half-day walk.
Lanhydrock and Pendennis Castle both follow the standard heritage pattern — walk the parkland or ramparts with the dog, but the interiors are assistance dogs only. Given how much of the appeal at both is the setting, that's less of a loss than it sounds.
The Eden Project is the same idea in a different shape: the outdoor gardens are fine, the Biomes aren't. Your ticket lasts a year, so splitting it across two visits works well.
Newquay Zoo, Paradise Park and the Cornish Seal Sanctuary are all assistance dogs only, for the same animal-welfare reason. St Michael's Mount is unusually strict — not even the causeway.
The practical approach for a week with a dog: build the days around gardens and coast, and treat the zoos and museums as the trips where someone stays outside with the dog. Almost every "no" attraction has a dog-friendly walk or pub within a few minutes.
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