Yes, accents change when words are combined/conjugated/etc. I'm not sure if there are any truly sentence-level phenomena, but there is definitely more going on than just "words have the same accent all the time". The NHK dictionary does include a fair bit of information about these rules.
To take your examples -- here are some answers I got from consulting the dictionary quickly. (Linguists, please forgive my half-assed terminology and use of the dreaded LH representation.)
1) This is actually more complicated than it looks! The 日 and 昇る in 日が昇る are both unaccented, and the の after an unaccented noun does not cause a drop, so the whole phrase has no accent. However, the 日 in あの日は (assuming you mean "that day" and not "that sun" IS accented. They are not actually homophones if you take pitch into account. あの is unaccented, and は coming after an accented mora = no accent, so あの日は = LHHL.
2) あの is unaccented as you say, but in combination with certain nouns, the whole phrase takes on an accent. I don't know how to account for this, but the NHK dictionary gives あの人 its own entry, with two possible patterns: LHLL or LHHH(drop) -- so maybe it is partly lexicalized, or at least the result of some rule obscure enough to justify including the result as its own dictionary entry.
3) When の is combined with a two-syllable word with a drop after it (like 橋 as you say), the drop generally "disappears" and the result becomes unaccented. So 橋 is LH(drop) but 橋の is LHH(no drop). Why 次の should work differently is a mystery. (The dictionary lists 次の間 as an independent item with either LHLL or LHHH(no drop), so clearly it is a known phenomenon.) -- Although, it should be noted that の working this way is kind of an exception in the case of noun + particle, which usually doesn't change the accent on the noun.
The problem with the information in the NHK dictionary (I mean the general "rules" rather than the entries for individual words) is that although there is a lot of it, it is for reference rather than pedagogical purposes -- and it's not even indexed very well. So it's not easy to learn from it if what you're after is a general survey of how Japanese accent works. I imagine it works much better if you're using the book as a reference while taking a course on how to speak NHK-style.