Are there any rough guidelines regarding whether a particle can be omitted or not?
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It's not really that で is omitted, but うち is diverted to something like "conjunction" or "preposition" in English. It can lead a full sentence ("sentence (終止形) + うち") or a noun phrase ("NP + のうち") to make an adverbial clause (sentence adverb) that means "in the course of; within". Sometimes, you can reword them using plain noun うち.
"adverbial" うち vs. "ordinary" うち
学校に通ううち、嫌でなくなってきた。 = 学校に通ううちに、嫌でなくなってきた。
三人のうち一番背が高い。 = 三人のうちで一番背が高い。
乗客のうち3名死亡、5名負傷 = (no equivalent)
(no equivalent) = 城壁のうちを歩いて点検する。
Many of other "formal nouns" (形式名詞) that refer to place or time, provide similar usage; なか "in/amidst ...", うえ "as well as ...", もと "under ...", とき "when ...", ところ "despite ..." etc., which more or less sound more high register or formal than the noun + postposition form equivalent (if any).
Some ordinary nouns (結果, あげく, 瞬間 ...) are gradually acquiring such usages, according to this recent research.
It's all a question of necessity.
In 「三人の中で」, で is expressing not a physical location so much, but rather a mental map of where something lies. In the case of 「ここで」, as ここ is a tangible (albeit relative) place, particles aren't be dropped after it, no matter what they are.
The rules vary from particle to particle.
を can be dropped in nearly every case where it's showing the receiver of an action.
It can't be dropped in cases of showing movement
が can usually be dropped unless it's important so that the sentence makes sense
This above sentence NEEDS が because grammatically the rules require it when questions contain が. In most other cases, though, it's okay to drop
に (and by extension へ) is usually kept in all cases, because it's pretty important in showing the direction of travel for people and/or objects.
仕事に行きました or 仕事へ行きました
In reference to your initial question, で is usually kept if it shows where something happened (i.e. where you waited) just as に is kept for travel. But, in the case of showing that from a selection, x is the most y, it's usually okay to drop, because it's not giving an indication of physical location. You also need to keep で if you're saying that you did something a certain way e.g.:
The rules are a bit more awkward than this at times, but these cover most things. You'll always be better off just listening to Japanese being spoken and hearing what other people do and what sounds the most "natural", which you eventually get a good sense of at some stage in your learning : )