"I like watching men's soccer."
Translation:Я люблю смотреть мужской футбол.
I am assuming that in Russian, as in most other languages, the "men's" part is usually left out in normal conversation as it is assumed to be the case unless specified otherwise?
There are a few adjectives that end in -ой instead of -ый in nominative (and accusative) singular. Maybe that is what's confusing you? Other examples are большой and плохой. I'm not completely sure, but I think all of these are end-stressed.
The adjective agrees with the noun it describes in gender, number and case: the noun "футбол" is masculine singular accusative, so the adjective "мужской" should be masculine singular accusative as well.
Doesn't the letter ж give a /dj/ sound? I don't hear it in "мужской." I would think it should be spelled "мушкой."
No, it's like the 'zh 'sound in 'pleasure '. My Russian teacher years ago always used to greet me as ' Джон ', with a fearsome ' D ' at the start, not the anaemic English ' John '.