(I)
The chicken crossed the road [to get to the other side]
Rex lit a fire [to keep warm]
Rex lit the fire [to keep warm]
[PROi To keep warm], Rexi lit a fire.
(NB: PRO in this example is controlled by Rex, so it must be c-commanded by Rex, hence the infinitive must have moved.)
(These purpose infinitives are adjuncts; they can be left out without causing ungrammatical.)

(II)
John hired Mary [to please Tom]
Reading 1: John hoped that [his hiring Mary] would please Tom.
Reading 2: [Pleasing Mary] is what John hired Mary for.

[To please Tom], John hired Mary
John hired Mary [in order to please Tom]

(NB: The first sentence is semantically ambiguous, but the second and third are not.)

Mary: I don't want [PRO to please Tom]
John: (But) It's what I hired you for.
(But) It's[ whati I hired you [to do ei] ]
?What is it [John hired Mary for __]?
What is it [John hired Mary to do __?
John hired Mary [to do what]?
What did John hire Mary [to do]

It was [to please Tom] that John hired Mary

Tell him [to call me back]
Tell me [(that) I'm not here]
Tell him (the answer/the information
(The complement of VP is here an argument; and note that it can't be fronted:)
*[To call me back], tell him (Arg) 
*[(That) I'm not here], tell him (Arg)

You are [to take the pills twice a day]
*[To take the pills twice a day], you are (Arg. of "is")
[imperitive:] (You will) Take the pills twice a day
You should take the pills twice a day
I told you [to take the pills], and [take the pills] you will

Who's [to know]?
Who would know?
*[To know], who is? (Arg. of "is")

Who am I [to say]?
Who am I [to judge]? (Arg. of "is")
How could I judge?

It is [to die for] (Arg. of "is")
You will die for it (?)

I'm happy [that I'm alive]
I'm happy [to be alive]
I take pills [to lower my blood pressure]
*[To be alive] I'm happy
[To lower my blood pressure] I take pills

I'm willing [PRO to talk with her]
*[PRO to talk with her], I'm willing
I'm hoping [PRO to talk with her]
*[PRO to talk with her], I'm hoping 
The infinitives are arguments of the matrix verb ("willing" and "hoping").

[  [IP He's [ NP the man [ NP [CP __[PRO to talk to __] ]
*[To talk to] he is the man (Subjacency violation)
He's the man [to do the job]
That's the score to beat
He's not much to look at

Passives:
He is supposed [to be at home]
( is supposed [he to be home] )
He is assumed [to be honest]
( is assumed [he to be honest] )
It is assumed [that he is honest]
He is believed by everyone [to be honest]
It is believed by everyone [that he is honest]

I don't know [CP who [PRO to talk to ___] ] 
*Who don't you know [ __ [PRO to talk to ___] ]

So: the embedded CP must be embedded w/in an NP
I don't know [the x such that I should talk to x]

"For":

What did you do that for?
I need something [___i [PRO to live for__ i] ]
I need something [thati  [ John will like ___i ] ]

? For what reason did you say that
*What reason did you say that for?
There is no reason [PRO to say that]
There is no reason [for [saying that]]
There is no reason [for [you to say that]]

[Burning your paycheck] would be foolish
[To burn your paycheck] would be foolish
[It] would be foolish [to burn your paycheck]
?[It] would be foolish [burning your paycheck]

"Reason":
Seems to have a mass usage:

There is little reason [to worry]
He did it without much reason
You have good reason to be concerned
It is not clear how much reason there is for concern
You don't realize how little reason there is for concern

(Likewise with justification, evidence, proof, support, cause.  These are all favoring relations -- what you used to call "grounding relations".)