Thank you @Belthazar & @Syphus so much for taking the time and further explaining this to me! 
Feels good learn about all of that and understand it! 
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聞 can in fact be read as もん, but the only modern word I can think of that uses it is 聴聞. 問 is also 口, I think those are the only two, not counting stuff like 們.
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The Kanji Study app says that it is broken down into the ground + slide + stick radicals.
Can you post a screenshot? I don’t see that at all. 
Just out of curiosity, are you using an iPhone? I know there’s an iOS version, but the creator has stated that he made it first, then switched over to Android to make that version and then left the iOS version behind and has no plans to bring them up to contain the same features.
If you’re on Android… I dunno. I guess, what version are you using?
(I realize this is probably possible to know from looking at your screenshot, but I don’t know anything about that kind of stuff).
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On android at least you have to activate it in the settings. On that screen, tap the three dots in the top right and chose settings. Then in the Details section check Radicals.
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My suggestion: get handwriting input for Japanese if you can. That will make things easier (though you might want to learn a little about kanji stroke order to facilitate writing and creating the right shapes), and honestly, that’s what I use when I run into something I can’t pronounce in Chinese or Japanese. Also, while I’m not sure how different IMEs work, try and see if you can get one that sorts kanji by the number of additional strokes they have after the radical. Some dictionaries do that, which makes it easier to find an unknown kanji.
EDIT: I see that a lot of people gave you far more detailed and helpful answers, so mine is a little… 余計. Oops. My apologies. Well, in any case, I’ll just leave this here in case you start coming across stuff that’s hard to look up in a dictionary with just radicals and stroke numbers. I don’t known of any Japanese IMEs with handwriting input (I typically use my Chinese keyboard instead), but I’m sure they exist.
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Great thread… learned a lot. Like OP, I had been thinking of radicals in the way that WK presents them, and didn’t know all the info about the actual radical system.
I have wondered for a long time why I struggle so much to find the kanji I’m looking for when I’m typing on a Macbook in hiragana, go to the 部首 tab to find the kanji I mean, and cannot find it under one of the “radicals” I thought it should be under.
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I still really appreciate hearing all the new ways people have given to search for the kanji. I have learned a lot of new tricks so thanks for adding yours! 
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One of the easier ways I found to search for Kanji is by drawing it on Google Translate. You can download the app onto your phone or use the website.
I’ve been wondering about thepaper dictionary radical lookup system. It seems very difficult. I assume that the reason you would use a dictionary is because you have encountered a kanji that you have never seen before. How are you supposed to guess which one is the correct radical to search for?
worst case you guess wrong and then have to try another one…
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Completely true, but in that case, the kanji dictionary’s search system doesn’t feel much better than a paper dictionary’s. I used to use dictionaries like that for Chinese, but that was literally about 10 years ago, and there were at least two classification systems (pinyin or radical + number of strokes overall, I think). Still, I guess you can’t complain if the kanji dictionary provides lots of useful information.
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If at first you don’t succeed…
Interestingly, my paper dictionary (Spahn and Hadamitzky The Learner’s Japanese Kanji Dictionary) has 聞listed under 門 rather than 耳, so I guess they’ve aimed for more intuitive over more accurate.
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As others said, you just try until you find the right one. The only issue a beginner might run into is if it’s something like 怪, where the left-hand radical is likely to be listed under 心 in some dictionaries, which has a different number of strokes.
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