Some core questions, indeed....
About budgets and such: Red Adair ( famous oil well firefigher) is quoted as saying, " You can have it cheap, fast, or good; pick any two." Think about it: cheap and fast is probably not going to be very good, fast and good is gonna be expensive, and so on. People planning a project for whomever is paying need to present the choice directly: what is most important? deadline? quality? budget? A sponsor who says they want all three needs to be educated in expectations management.
As for a deadline that is going to be missed, the most common choice is deliver less or get approval to extend the deadline. This again is the sponsor's choice. They may also elect to throw more money/resources at the project, which can help until the point of diminishing returms (the "mythical man-month" and all that). For me, the key is to address the problem as soon as it comes up, don't wait hoping things will improve somehow. Bad news now is much better than horrible news later.
As for the conflict management questions, I will defer to others who may have a more deft touch in these matters, but if I am really faced with it and I am not in charge of all the people involved, I escalate the problem til it gets to some one who is, they probably make a whole lot more money to deal with situations like that.