I've seen 浮かべる used thus: 「してやったり！」の表情を浮かべる and 不敵な笑みを浮かべる. Given that it uses を with a noun, it would be a transitive verb, right? I'm wondering if the best translation would be "To show/to express". Finally, is it used with only facial expressions?
As for the rest of them, I'm struggling to properly distinguish them. I would like to be able to say "It makes me think of home" or something along those lines. If these words are uncommon or there are better words, then please let me know.
The obvious difference here is that 浮かぶ is intransitive whereas 浮かべる is transitive. Therefore you have to say 彼は笑みを浮かべた and 彼の顔に笑みが浮かんだ. The former implies he did it deliberately, while the latter implies the smile might have appeared beyond his control. 浮かぶ/浮かべる can be used with various things that can "surface" or "emerge". For example 皮膚に浮かんだ血管 refers to a vessel that has come to the surface of the skin. 容疑者として浮かんだ人 refers to a person who emerged as a crime suspect during an investigation.
思い浮かぶ (intransitive) and 思い浮かべる (transitive) are both used to express ideas in one's mind, as the 思い part suggests. 浮かぶ is interchangeable with 思い浮かぶ when it express some idea coming into your mind suddenly or spontaneously:
- ある考えが思い浮かんだ。 A certain idea came into my mind.
- ある考えが浮かんだ。 A certain idea came into my mind.
思い浮かべる can be simply translated as "to imagine". It's usually not interchangeable with 浮かべる.
- 星を見ているところを思い浮かべてください。 Imagine you're watching stars.
- [×] 星を見ているところを浮かべてください。 (incorrect)
As for your example, It makes me think of home.
It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ.
On the other hand, 故郷が浮かぶ sounds like images just simply pop up in your head and it's not so important or you have a vague image of hometown. Since 浮かぶ is associated with an expression like 雑念が浮かぶ: miscellaneous thought comes into mind. which is a collection of unstable images. Therefore, in my opinion, 故郷が思い浮かぶ sounds you are retrospecting your hometown rather than saying 故郷が浮かぶ.
