***NOT A TEACHER***
dawnngcm, good morning.
(1) I am disappointed that no one else has yet answered all of your questions.
(2) May I make a respectful suggestion: if you submit a short post with only one question, maybe more people will reply.
(3) I searched all my books and checked out the Web. I am NOT satisfied with my findings, but here is what I learned.
(4) You say that the combination "verb + to-infinitive + adjective" is "rare."
(5) I had never thought about that matter before. You may be right when it comes to adverbial phrases of purpose ending in an adjective.
(6) I found only one -- in my favorite grammar book: He struggled TO GET FREE. The book says it expresses "mental direction" or "purpose."
(7) It is similar to your "I learn to be smart" or "I study (in order) to become smart." I feel that is an adverbial phrase: it explains your purpose for studying.
(8) But remember, please: adverbial infinitive phrases are very common -- but NOT those that end in an adjective. I couldn't find any other examples!!! Of course, they must exist.
(9) When it comes to adjectival phrases, however, there many many examples. In other words, they are far from being "rare."
(10) The news proved/turned out to be true./He lived to be old./The weather continues to be pleasant./He seems to be all right./ The children appear to be asleep./ His ambition is to become rich.
(11) They seem to often appear in passive sentences: He is said to be good./ He was believed to be rich./ He is said to be quiet./The traffic was reported to be heavy./Her father was presumed dead.
(12) You asked how the combination "verb + to-infinitive + adjective" can sometimes be adverbial and sometimes adjectival. I have no answer. Perhaps you can submit a similar question in another post.
Good luck. I'm sure you will soon be speaking English very well.