Yes, the name is more inspired by the modern Camus interpretation than the ancient one.

Each trigger is separate, so you can practice them separately and their effectiveness may develop separately. In fact, you can have unconnected triggers for isolated situations ("When the dream is destabilizing, I will remain still and attempt to DEILD"). They become connected because the events happen in sequence, not because you practice them together or in which order you recite them. In other words, you form the chain by choosing events that happen in sequence. But once you've chosen the events, you can focus on the "links" on the chain individually. Naturally, it makes sense to emphasize the first trigger (induction) because the rest depend on it. But I think it's fine to start adding more even as you're ingraining the first. As you proceed, you might discover "gaps" in the chain which you can work to adjust or fill in.

To me, the triggers are often just a firm statement to remind myself of something, stated just once or a few times, not necessarily a mantra to be repeated endlessly. Some are more like a checklist. Some are phrased as a question, to cue a response ("What is my goal for tonight? Fly to the moon"). I think the induction trigger is the only one I repeat like a mantra ("The next time I am dreaming, I want to remember to recognize that I am dreaming"). Thinking back on my own practice, sometimes I would recite things in order, sometimes in reverse order, and sometimes in no particular order. Reciting things in reverse order might help when you have some end goal you want to work backwards from. Or reserve order might help so that the the earliest events are recall last, meaning they are more recent and salient in your mind.

To your second question, the effectiveness of a trigger depends on the strength of memory association with the trigger condition. And you can roughly estimate the strength of a memory association with the frequency of recurrence in your dreams. That's why, when looking for dreamsigns, you look for the things that appear most frequently in your dreams. "The next time I see something odd..." seems a bit too inspecific to me. But if you frequently find yourself in a dream thinking "Hmm, that's odd" then it might be effective. You want to be very very particular with the words you choose so that the memory association is as strong as possible. In that example, if you would naturally choose the word "strange" instead of "odd," that could make a big difference. A better example: Suppose seeing your mother is a dreamsign. But in your thoughts and words, you might usually call her "Mom." In that case, "The next time I see Mom..." is stronger than "The next time I see my mother..." because it resonates with your personal memories and habits.

That might have been long winded. Shorter version: Triggers are reminders. They are most effective when they are strongly associated with your personal memory. If the events chain together, you can build more complex and reliable habits, but they can be useful in isolation too. They don't all have to be mantras. Write them all down as a checklist and recite them like that. Maybe one checklist for before bed, another checklist for MILD, another for WILD, etc. Choose a few of the more important items on each checklist to repeat several times for emphasis. Choose just one or two to use as mantras to repeat until you fall asleep. Analyze, adjust, and refine as each experience adds new information.