Learn English – “Reply him” or “reply to him”

What is it called when the preposition is omitted?  It's called an indirect object. 

That's what an indirect object is.  An indirect object is licensed and governed by a verb.  When a preposition is involved, the object is a prepositional object or an oblique object. 

Oddly enough, the cited dictionary page doesn't manage to offer a clear example of this verb's ditransitive use.  The example "he was awarded the Military Cross" is cast in the passive voice.  The ditransitive construction is more obvious in the active voice: "They awarded him the Military Cross."  The example "a 3.5 per cent pay rise was awarded to staff" is not a ditransitive example, even when cast in the active.  "Management awarded a 3.5% raise to the staff" includes a direct object and a prepositional object, but there is no indirect object in sight. 

You don't seem to be confused by the ditransitive use of this verb in the active voice.  However, passive-voice participles do seem to confuse you. 

Such confusion is easy to understand.  For most verbs in English, the past-tense forms and the so-called past participle forms are identical.  Even the "past participle" label is confusing, given that participles have no tense. 

"The Academy awarded the Oscar" looks quite similar to "the actor awarded the Oscar".  Without further context, they both can be parsed as clauses in the active voice and past tense.  However, "the actor awarded the Oscar has refused to accept it" requires a different parsing.  Here, "awarded" is a participle, "awarded the Oscar" is a participial phrase which modifies "actor", and "the actor awarded the Oscar" serves as the complete subject of "has refused to accept it". 

You're not the only one experiencing such confusion.  It's nearly universal.  Sentences like "the actor awarded the Oscar has refused to accept it" are garden-path sentences.  It is easy for anyone to assume that "awarded" is a finite verb until the phrase "has refused" is encountered.  Since "has refused" is finite and must have a subject, we have no option but to re-parse "awarded" as non-finite and without subject. 

The examples that you provided are not garden-path sentences.  Each one contains a clue to the participial nature of the verb in question before it is encountered: 

The phrase "on the plaque awarded her" has "on", a preposition which licenses an object.  The phrase "the plaque" cannot serve as the subject of "awarded" since it serves as (a part of) the object of "on".  A similar explanation serves for the prepositional phrases "about the Nobel Prize awarded Ernest Hemingway" and "by the Oscar awarded Michael Radford's Il postino . . .".  In "who refused the Oscar awarded him", "the Oscar" is (a part of) the direct object of "refused".  In the absence of a subject, there is no clause and there is no reason to assume that the verb's form is finite. 

Well, your assertions are not entirely correct. For example it's fine to say

The doctor asked the history of my illness

Use "about" when you want to know the general facts around some information, or when the direct and indirect object of the verb might not be clear. Example:

He asked the computer.

He asked about the computer

In the first example, the computer is the indirect object, from which he expects an answer. In the second it's the direct object, the subject of his query.

In the same way, it would be wrong to say something like, "He asked my mother's health," because that makes no sense. You can't ask anything of someone's health. Instead say:

He was always very friendly and often asked about my mother's health

Because adding "about" can imply that the information is doubtful or unusual, don't use it when requesting specific, well-defined information such as, "The doctor asked my date of birth." Instead use it when you want to know more peripheral detail:

The doctor asked about my date of birth -- whether there was an official record, and if I was sure it was the correct date. I think he was trying to say I might be younger than I think I am.

I asked about her name, and what it means in her own language, but she told me it actually doesn't mean anything. Her parents just liked the sound of it.

In some situations, however, "about" implies courtesy, as indirect questions may be more polite in certain contexts.

The customer asked the price of the car.

The customer asked about the price of the car.

Note there are many other ways to express this kind of polite deference:

The hesitant young man inquired into the price of the ring, and whether he might possibly pay in installments.