Yes: they necessarily do so.

Despite the article's statement, tardigrades are not extremophiles, they can't live in extreme environments. What they can do is exist/survive in hard vacuum and other extreme environment by entering a super-hybernation mode (cryptobiosis).

Furthermore while they can survive fairly long spans in extreme environments they're not actually immortal, the FOTON-M3 mission resulted in 32% mortality after 10 days in hard vacuum but protected from UV (the survival rates were much lower for those exposed to UVs).

yes because there is no water, air, or food on the moon for a suitable habitat for tardigrade reproduction

We can sequence its DNA and see how similar it is to the ones on Earth. If there are living organisms on the moon that share a common ancestor with Earth tardigrades then they diverged a very long time ago and their DNA will reflect that. But if their DNA is identical to those on Earth then we know they came from Earth.

Do you know how much tardigrades DNA have changed since they first appeared on earth?

I do not know exactly how much but this is something that can be approximately quantified.

yes, these tardigrades were sent to the Moon in a rocket