Is kimchi a suitable addition to a noodle soup?

Kimchi are not preferred to be added directly to soup. Usually you would fry the Kimchi (ideally in Sesame oil) to slightly transparent or having minor brown on the edges, before adding into the soup, and boil for . That is not only applicable to KimChi JiGae- but also other varieties of KimChi broth (e.g. hotpot, Kimchi ramen/noodles (BuDae JiGae) etc.)

Adding Kimchi directly to soup/broth and boil is more a Northern Chinese way of cooking (where Kimchi is not called kimchi but literally called "Spicy Pickled Cabbage").

Frying beforehand and adding directly to soup/water can taste very, very different.

Yes. It's almost a required element for me any time I'm eating the standard cheap ramen or noodle bowls. I usually add it at the end so I don't cook all the crunch out of the kimchee.

I often take five or six frozen gyoza/dumplings, add them to the small pot of water, bring that to a boil, then add the ramen noodles, then, after the noodle cooking time, add some of that salty seasoning packet and the kimchee.

As pointed out in the selected answer, cooking it, as a step for soups and stews is also very standard, but I'd disagree about it not also being a fine addition, as is, to a noodle soup.

Not at all. Heres a recipe you could try http://norecipes.com/blog/kimchi-jigae-recipe-kimchi-soup/

Absent any information (not even a picture?) I'd assume it's the common variety with the dominant seasoning being spicy red chili. It's certainly a reasonable thing to put in a soup: kimchi soup and kimchi stew are popular dishes. It's also a very common standard side dish. If you're unsure, just open the can, taste it, and see what you think you'll like best.