Second, they didn’t carry out the mission on the Moon. The purpose of the mission was to land on the Moon, so the “on the Moon” location applies only to the purpose of the mission, not the carrying out of the mission. They carried out the mission from the launch from the surface of the Earth, to landing in the ocean and everywhere along the long, loopy path in between. The mission was not complete until the astronauts were safe at home and the moon rocks and photographic film were delivered to the appropriate entities on Earth.

That's not what this sentence says. It's talking about a SaqmeH Qu' landing mission, not the entire mission of going, landing, exploring, and returning.

Thirdly, {maS SaqmeH Qu’} is ungrammatical because you can’t put a verb in the middle of a noun-noun possessive construction.

Says who? SaqmeH Qu' is a noun phrase, and noun phrases can count as the "noun" parts of a noun-noun construction. Are you saying that if I have a pe'meH taj cutting knife, I can't talk about the jonwI' pe'meH taj engineer's cutting knife?

I have no problem with {maSDaq SaqmeH Qu’} meaning “The mission which has the purpose of [something] landing on the moon. It is a little ambiguous because without a stated subject for {Saq}, you could interpret it as “In order that the mission lands on the Moon…”, but then we’d have to find something later to apply this purpose to, and there aren’t any good candidates. Besides, how does a mission land on the Moon?

We have lots of these "infinitive" purpose clauses. pe'meH taj cutting knife, ghojmeH taj boy's knife (learning knife), qIpmeH Qatlh'a' difficult to hit? QongmeH qItI'nga Duj  K'Tinga-class sleeper ship, qa'meH vIttlhegh replacement proverb. There is no trouble interpreting SaqmeH Qu' as landing mission and maSDaq SaqmeH Qu' as moon-landing mission.

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SuStel
http://trimboli.name