1. Does self-awareness coax in dreams by means of a habit(making it second nature). I know this is a question you just talked about, but if the memory responsible for such maintainance of a habit seems to be linked to intention. I will explain: Not all habits coax to dreams, but habits that we use intention to support seem to coax well
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Intention and habit are opposite (or at least different) things. An habit is something you do unconsciously, or at least a behavior that is integrated on your daily events. Intention implies a conscious attitude towards something. I get what you're trying to say though, and I honestly don't know what makes the "intent" influencing the "habit" in dreams: my view is that it's prospective memory that is helping you remembering something. Prospective memory can still take effect under distraction or temporal distance between the reminder and the cue, especially if we're talking about event-based prospective memory. In simple terms:

- While you develop the habit, prospective memory takes a larger role (you need to "work for lucidity);
- After the habit get's implemented, it still won't happen all the time (I can't comment on lucidity like hukif's/sageous/sivanson in here), but you need less reminders. I got no clue why, in this particular aspect I'm just speculating, so other theories would be welcomed

2. When making self-awareness a habit and using intention to support it, does it actually coax to the dream(has there been direct studies to show that?)
LouaiB: the (I think) most recent meta-analysis on lucid dreaming technique effectiveness highlighted one thing: Poor methodological quality of all studies regarding EVERY technique listed. Yes, that includes LaBerge's MILD. Guess that should give you an idea of what the answer for that question is xD