I'm not too sure what "left out" emboldened above actually means. Does it refer to sentences ending with the te form? I guess so
As toritoribeさん and the wikipedia page explain, the ~て form can always be thought of as connecting one clause to another, but the following clause to which it's connecting can be omitted in casual speech, especially when whatever would follow is clearly implied from context.
Your original example of ごめん、待たせてしまって! is an inversion, rather than a strict omission, but you can also find cases where the second clause isn't explicitly stated at all, because it's implied.
For example,
A: 仕事帰りに一杯飲みに行きませんか? Shall we go out for a drink after work?
B: すみません!今日はちょっと用事があって・・・ Sorry! I have plans today, so... [I can't go].
The trailing off after the ~て form is softer/less direct than explicitly saying 今日はちょっと用事があって行けません, ちょっと用事があってダメなんですよ, and therefore is quite common in everyday Japanese conversation.
The "left off" emboldened in the second sentence is not clear to me either. Does it mean both くれる kureru and 下さい kudasai are omitted?
"Left off" in the second sentence is the same thing. You see the examples of 食べて/読んで (or 食べてね/読んでね, which you'll often find in everyday speech) can serve as requests without explicitly including a verb of giving.
The example of "殺してくれ" has the くれ in imperative form, hasn't it? And the imperative form is stronger than the one ended with te form, isn't it?
Yes, くれ is an imperative form and thus stronger and more than ~て、~てね、~てください, etc. This particular sentence sounds like it could come out of a comic book, video game, war movie, or what-have-you.