Translation:I came to Japan this month.
I'm not sure how it is that those both sound like "iki" to you. However, there is something I can add that you might find will help.
The pitch accents for 来ます and 行きます mean there's a difference in pitch for the "き" sound in those two words.
来ます = LHL. (Meaning: き = low, ま = high, and す is low.)
行きます = LHHL. (Meaning: い = low, き = high, ま = high, and す = low.)
So, in spoken conversation, the two should also sound different in terms of pitch contour.
今 月 is certainly a noun. は is the topic marker indicating that, whatever happens in the rest of the sentence, it happened with regards to 今 月 (this month). Once we've established the topic (the timeframe of this month) we can discuss the action taken: the subject coming to Japan (日本 に 来ました ) which happens to involve another noun (日本 ) at the beginning of the phrase. Therefore, we have two nouns with a Particle between them
That's true. However, I think it should be pointed out that in Japanese, you only use verb 来る when the motion happens towards the speaker.
Thus, 今月、日本に来ます implies that the speaker is already in Japan and is perhaps talking about someone else's coming to Japan, or maybe they meant to say また日本に来ます "I'll come again (back) to Japan" so they're going on a trip less than one month long.
Wrong. 今 in 今日 doesn't have a separate pronunciation from 日. The whole set is read as きょう. This happens when they applied Chinese writings on Japanese words. Kyō was "today" before the writing system was imported. They just put the Chinese writing of jīnrì (今日) on it because they have the same meaning. Although, in some situations, 今日 is actually read by the reading of each symbol. In this case 今日 is pronounced こん・にち (that's where こんにちは comes from. "今日は")
The 今 have the いま, コン pronunciation like 今 ( ima / now ), 今月 (kon getsu / this month ) but a lot of kanjis have a special pronounce in some situations, 今日 ( kyou / today ) is one of then Normally 2 kanjis together combine their pronunciation added, the 今日 is a exception. (example: 時間 / Hour > JI + KAN, 勉強 / Study > BEN + KYOU)
Because the verb is "come" 来る (くる) conjugated to polite past tense "came" 来ました (きました).
It's not the verb "go" 行く (いく) conjugated to polite past tense "went" 行きました (いきました).
The speaker is in Japan at the time of saying this sentence. So it's not "went", which would instead be used if the speaker were in another country talking about a previous trip to Japan. ^^