Wikipedia explains this in depth, including a relatively comprehensive list of uses of infinitives in English, but here's a compact answer.
Infinitives do act as verbs - forming infinitive verb phrases
Infinitives are by definition non-finite, generally do not have a subject, and so form non-finite clauses. These may be used as nouns, adverbs, etc. The ...
The are called "infinitive" because they don't have inflections for tense or person. They use the bare form of the verb.
The verb forms called "auxiliary" are contrasted with "lexical" verbs. Using the verb from your example, the word "sing" is a lexical verb. When used alone, it changes with tense and person:
The infinitives give the purpose of the studying, and and can be paraphrased with "in order to"
I studied [in order] to be a doctor.
I studied [in order] to pass the test.
The infinitive phrases are not the object of the verb "study" but form an adverbial modifier. Above the object is implied, but you could give a noun as the object.
refuse in this context means to express a determination not to (do something).
Therefore the structure refuse to + infinitive is needed. (e.g. I refuse to go to school)
For more information on this structure and on the usage of gerund have a look at this:
The first example is a list of three actions. The order of the items in the list doesn't matter. The list could be "lifted weights, jumped rope" and it would still mean the same thing.
The other two examples are narratives describing a sequence of events. Order does matter. The final event (gave the gift, put a pillow) may be more directly related ...
You are confusing "meaning" with "grammar". As you correctly observe "bid" and "order" have a similar meaning in this context, but different grammar. There is no strong reason why.
Note that this sense of bid is now dated, or even archaic, and so shouldn't be used except in fixed expressions like "bade farewell&...
The World Trade Center is, famously, the complex of buildings destroyed in the 9/11 attacks and now rebuilt. However, you could say China is close to becoming the centre/center of world trade.
We would not use the infinitive in sentences like this.
It is a sentence with the coordinating conjuction but, where an independent clause "I have no choice" may join either the phrase to run away or running away.
Because the object in the independent clause is choice that is a noun and the conjunction "but" is a coordinating one, we should use a phrase functioning as a noun after but.
It may ...
As Kate said, there is no difference in meaning between the two. Sentence #2 is an example of elision, where a word ("to") has been omitted because it is understood from the context.
Elision is common in tight prose, which might be subjectively better, but it is not more "correct". If memory serves, Strunk and White even has a section on ...