Rainiers makes the most sense.
What do the others have to do with Washington/Seattle?
Not sure if serious and someone from Seattle can probably answer this better than I can but
1. Cougars - Cougars are common in the Pacific NW (and some other parts of western USA, Canada, and Mexico). Plus, maybe it will be a draw to Washington State fans, since the cougar is their mascot.
"Today viable, breeding cougar populations are found in just the sixteen states of Washington, Oregon, California, Nevada, Arizona, Utah, Idaho, Montana, Wyoming, Colorado, Nebraska, New Mexico, South Dakota, North Dakota, Texas, and Florida. In most of these states,
cougars are the only species of native large carnivore that survived extermination programs."
State by State
2. Eagle - SAA except eagles are in more states. Still, they're relatively common in the Seattle area.
Bald Eagle Facts - NatureMapping
3. Emeralds - Seattle is the Emerald City bc of all the rain. It's been a nickname for at least decades.
4. Evergreens - Lots of trees there. The whole early Pacific NW economy was based on logging and Seattle was a mainly logging town. It's a not only a nod to their current ecology but also their history.
History of Seattle before 1900 - Wikipedia
5, 6 & 7. Sea lions ,seals, and whales. See #2
8. Sockeyes - They range from the Columbia River in Washington/Oregon border (and very rarely down to Northern California) to the Canadian Artic/Siberia and then down to Japan in the Western Pacific. Although it has a fairly large range, the number of highly populated areas in NA where they're common is very limited. Seattle (and Vancouver) would be one of the very few cities where naming a team the Sockeyes would make sense (from the perspective of actually having those animals in that area).
Sockeye salmon - Wikipedia
Raniers make sense, too, as you said.