We have been gifted a second hand cot mattress which has some visible stains on it. I note that you specify that a safe mattress should be 'clean'. Should we purchase a new mattress?

Red Nose has the following information

Any previously used mattress should have been well stored, firm and clean. If the mattress is damp, torn or mouldy or there is any doubt about its safety, including fit, it should not be used.
https://rednose.org.au/article/is-it-safe-to-use-a-second-hand-mattress

The presence of stains on the mattress would indicate that it may need to be replaced.

Red Nose has the following information about second-hand products.
https://rednose.org.au/article/second-hand-nursery-furniture