First off, Sim just gave a ton of very sound advice.
A few things to add on my part:
I read on Wikipedia that Quanzhou has a third pronunciation they call "vulgar pronunciation". Anyone has any clues if there's such a thing in Taiwanese too? And if not, how useful could it be learning them from Quanzhou sources?
No idea what they're talking about. Can't trust everything U read on Wikipedia, U know.
Feel free to post a link to the article. And, whatever it is, I'm pretty sure that if it exists in Coanciu, it'll also exist in Taiwanese.
Coanciu sources could work great, as long as U don't confuse yourself. There are also two or three different types of Coanciu dialects, and they're not esp. similar even in comparison to Ciangciu, if I'm not mistaken. For a linguist, and it seems like U are one, it would be relatively easy to go from Coanciu to the rest of Hokkien, and relatively hard to go the other way. For one thing, all 8 tone classes have been kept intact in Coanciu, at least in the city of Coanciu. One more caveat, though, is that the dialect of Coanciu has been changing fast, so sources from different eras may contradict each other.
Very useful 字典, based on Coanciu Hoklo:
http://solution.cs.ucla.edu/~jinbo/dzl/
1. Why sometimes 台語辭典(台日大辭典台語譯本)查詢 give weird pronunciations? (I marked those with a "?" and bold)
Let me address 香 = "phang" first. Phang is the Hoklo word for FRAGRANT. The "etymologically true" kanji is 芳, AFAIK. Meanwhile, the word (by def., written word) for FRAGRANT in Classic Literary Chinese is 香. Meanwhile, 香 is also associated with the Hoklo etyma "hiuⁿ" and "hiong".
So, 香 = "hiong" would be analogous to 山 = "san" in the JPnese context, whereas 香 = "phang" would be analogous to 山 = "yama". The analogy isn't perfect, since "phang" is in itself Sinitic. The shared feature is that in both cases the kanji is not "etymologically correct".
As for the other examples, no idea. Hiō is the only other word here I even somewhat know. I suspect these are beyond the grasp of the avg under-60 TWnese Hoklophone. Then again, I'm not even that. My tentative advice (not fully informed, etc.) is that U should ignore those as being idiosyncracies in the way the 台日大 is set up. 台日大 is probably unsuitable for this purpose, BTW.
2. Where there's an e/i (or a/o) variation (tèng/tìng, chheⁿ/chhiⁿ, hiong/hiang, etc.) should I stick with only one?
Short answer yes, but no penalties for straying. Whatever is easiest for U.
Details: -iong/-iang is generally Coanciu vs Ciangciu... All -eⁿ endings go to -iⁿ in Coanciu and Amoy, but Ciangciu has -iⁿ endings as well, just not on 生, etc.
Hiuⁿ / hiaⁿ seems like it might be different layers. I don't know any words using 香 / hiaⁿ. C.f. niû and niâ 娘 though.
Chhéng / chhíng is an attempt to illustrate the quality of the vowel. Nobody does this consistently, so I'd ignore it. Standard POJ uses -eng, but U'll see -ieng spellings too. The TW MOE adopted -ing. Not phonologically significant.
3. I see a lot of tonal variations. Do I have to memorize which ones take one tone and which ones take the other tone?
Short answer: they are DIFFERENT WORDS that just happen to be etymologically related, and happen to be attached to a certain kanji as well.
The relationship between words and kanji in Hokkien is very similar to what goes on in Japanese.
Any possible explanation to why that happens?
Short answer: different eras. Lāu is old-school "original" Hoklo. Ló is a "Tang pronunciation", imported during the Tang or later. Láu is intermediate between the two. THESE THREE ETYMA HAVE DIFFERENT MEANINGS, CONNOTATIONS OR USES IN HOKLO.
The same thing happens in Vietnamese. Bốn, tư and tứ all mean FOUR, but they are different words with different uses. Tứ is the Tang pronunciation, what they call "Sino-Vietnamese". It's associated with 四. Tư is a Sino loan too, but an earlier one, and not considered "Sino-Vietnamese". Ba is the "native" Viet word. Bốn, and tư too I believe, are written with "exotic" Vietogenic kanji...
Another analogy: agua and the acua- in acuático are different etyma with different uses, but they are etymologically related and would be etymologically associated with the same kanji, if kanji were involved. Other, more breathtaking examples no doubt abound.
Hokkien literary vs. colloquial pronunciations are quite a distinctive part of Hokkien (as you and other regular Forum members undoubtedly know). Apparently, other Sinitic languages have it too*, but it's particularly prevalent in Hokkien.
True, within "Sinitic", they are particularly prevalent in the "Min" languages. They're also prevalent in Vietnamese (though analyzed differently, as just being different words) and in Japanese. The "Wu" languages also have this going on.
This all goes back to the layering of the Sinitic elements in these languages -- Hoklo, Hokciu, Hinghwa, Hainamese, Japanese, Vietnamese, etc.
These languages have layers of Sinitic that can be traced to different bygone eras. These eras usually represent either high-water marks of "Chinese" civilization or periods of chaos when people and scholars fled -- always south.
These areas (Hokkian, Japan, Vietnam) were so remote from the centers of Sino-culture, that they could be "Sinfluenced" from the center only when Sino-culture was at full flower, or through the movement of people, and in both cases this Sinfluence may've indirect and lagging. Meanwhile, most of what is today China Proper was closer to the centers of Sino-culture, and so Sinfluence took place on a continual basis ... so that layers did not form to the same extent that we see in VNmese, JPnese and Hoklo...