Is this topper thick enough to soften a hard mattress?

The 500 gsm cotton fill is on the denser side for a topper, so it adds noticeable softness. It is not a memory foam pad and will not change the feel of a very firm mattress dramatically.

Will it stay in place on the bed?

There are no straps or elastic corners described, so it sits on top of the mattress like a pad. Expect to straighten it, especially on a bed that gets a lot of movement.

Cotton toppers of this density are usually spot cleaned or washed cold on a gentle cycle and dried flat. Check the care label before washing, since a dense cotton fill can clump if machine dried on heat.

Cotton breathes better than polyester or foam, so it is a reasonable choice for warm sleepers. The dense fill still traps some warmth, so hot sleepers may prefer a thinner cotton pad.