Though you're correct that many of the pairs you've provided are similar semantically, you've confused the functions complement, object, and adjunct(traditional grammar's 'adverbial').
First off, you'll want to get form and function separated. The form of dependent clauses, and phrases, is determined by their internal structure: NPs are headed by nouns and allow a certain range of dependents; to-infinitival clauses are headed by a verb in plain form and allow a range of dependents different from those found in an NP. So, a to-infinitival clause will never be an NP, but it may appear in the same functional category in a larger structure.
Then, you can start to look at the function of these dependent clauses and phrases in clause or phrase structure.
She - is - likely [to - start - her new project]
Subject - predicator - predicative complement
She - may - [start - her new project]
Subject - predicator - catenative complement
In I, to start her new project is a to-infinitival clause functioning as dependent (specifically a complement) in the adjective phrase headed by likely. It is a complement because it is allowed only by likely and other adjectives allowing a to-infinitival clause as complement, not by adjectives in general: *dry to start her new project.
In II, start her new project is a plain infinitival clause functioning as direct dependent (specifically a complement) of the predicator may. The specific type of complement is a catenative complement. These are non-finite (plain infinitival, to-infinitival, gerund-participial, or past participial) clauses specifically allowed by the verb acting as predicator.
It is necessary to note here that the to-infinitivals in your examples are not adjuncts, but in fact complements in AdjPs in A and B, and catenative complements in C. This becomes clear when we note that in order may not be inserted while keeping the same meaning,
?I'm willing in order to eat that cake. (Willing to do what?)
I'm dying in order to eat that cake. (How terrible!)
*I hoped in order to go to the movies.
I begged in order to go to the movies. (What did you beg for/to do?)
*I long in order to get an A.
I volunteered in order to help the kids. (What did you volunteer
I was aiming in order to receive the trophy. (What were you aiming
Though there is some overlap between verbs that allow different sorts of complement - object, catenative complement, predicative complement - the complements themselves are best viewed as distinct categories each allowed by a specific set of verbs. For example, what follows get in the each of the sentences below is a distinct sort of complement:
Mark - got - a dog [object]
Mark - got - excited to go one the trip. [predicative complement]
Mark - got - me - a dog [indirect object - direct object]
Mark - got - chosen as goalie [catenative complement(past
Mark - got - to go first [catenative complement(to-infinitival
Mark - got - me - to go first [object - catenative
Though get allows these constructions, if we pick another verb that allows the first, say sniff, we'll quickly see that the others are not allowed.
Mark - sniffed - a dog [object]
*Mark - sniffed - excited to go on the trip. [predicative complement]
*Mark - sniffed - me - a dog [indirect object - direct object]
*Mark - sniffed - chosen as goalie [catenative complement(past
*Mark - sniffed - to go first [catenative complement(to-infinitival
*Mark - sniffed - me - to go first [object - catenative
Though the last two could be understood as adjunct of purpose in some strange context, they are not understood in a sense similar to the catenative complement examples with get.
Long story short, yes all of the pairs you provided have similar meanings, no their structures cannot be analyzed as equivalent.