When you tap Heimerdinger, do you activate the tapped abilities of all other cards you have, or only one?

You only activate one ability, not all of them.

Heimerdinger, Inventor gains copies of all exhaust abilities from friendly legends, units, and gear simultaneously — but when you exhaust him, you choose exactly one of those copied abilities to activate. You pay that one ability's cost (including any additional costs like energy), and it resolves. Since exhausting Heimerdinger is itself the cost to activate any of them, you can only activate one per exhaustion unless he gets readied again.

(FAQs 1044, 8927, 5639; Rule 377)