Cfrtjryk

I accept that my premise may be incorrect, but here it is.

The word alone, when used as an adjective, seems only to fit in sentences of the form:

1. The X is alone.

and not in the form:

2. The alone X...

I can't think of any other adjectives that have this property, that is, that are allowed in constructions like (1), but not in (2).

Question 0: Am I right in thinking that The alone X is not correct English?

Question 1: Are there any other such adjectives that have a similar behavior?

Question 2: Is there a word for such an adjective and is there any rationale for their existence?

There are three main positions for adjectives in English:

1. 
   

   Predicative: where they occur as the Predicative Complement of a verb, such as FEEL, LOOK, SEEM or BE:

   
   
   
   
   
   • He felt alive.
   
   
   • The elephants looked serene.
   
   
   • She seemed discombobulated.
   
   
   • He was irate.
   
   
   
   


2. 
   

   Attributive: where they occur as the modifier of a noun:

   
   
   
   
   
   • You're talking utter nonsense.
   
   
   • That's a large elephant you have there.
   
   
   • I can't stand the hypocritical cant of politicians
   
   
   • The once ubiquitous sparrow is now an endangered species.
   
   
   
   


3. 
   

   Postpositive: where they occur after the noun they are modifying:

   
   
   
   
   
   • the quickest route possible
   
   
   • the worst conditions imaginable
   
   
   • the best deal available
   
   
   • the body politic
     
   
   
   
   

Notice that the adjective responsible can occur in all three positions. So we can say that responsible can be a predicative, attributive and postpositive adjective. This would basically just be saying that responsible is an adjective that can undertake all the normal adjective functions that we normally expect adjectives to be able to.

However, some adjectives can only be used in some of these positions. In other words the syntactic functions that they can carry out are restricted. For example, notice that the adjective alive in group one cannot be used attributively:

• *I found an alive man. (ungrammatical)

Some people say therefore that alive is a predicative only adjective (this is not correct, as explained further below).

Conversely, the adjective utter in group two can only be used attributively:

• *Their stupidity was utter. (ungrammatical)

Utter is therefore often described as an attributive only adjective.

Most adjectives that occur postpositively in set phrases only occur postpositively, such as the word politic in group three above. We usually just say that the words are postpositive adjectives.

The Cambridge Grammar of the English Language (Huddleston & Pullum, 2002) broadly categorize adjectives with restricted functions into two categories: attributive-only adjectives and never-attributive adjectives. Attributive-only adjectives are:

• "Adjectives that do not normally occur except as (heads of) attributive modifiers [...]".

Never-attributive adjectives are:

• "Adjectives which can occur predicatively or postpositively, but not attributively".

Note that this is a much better description of the adjective alive, for example, which can of course occur postpositively as well as predicatively:

• Anything currently alive is salvageable.

References: The Cambridge Grammar of the English Language, Huddleston & Pullum, 2002. pp 553-562.

It is true that 'alone' can occur as both an adjective and an adverb. This is attested by many dictionaries.

If you look at the Oxford definition, you see:

alone

Having no one else present; on one’s own:

[AS PREDICATIVE ADJECTIVE]: 'she was alone that evening'

[AS ADVERB]: 'he lives alone'

Notice that Oxford classifies it as a predicative adjective in its example sentence. Predicative adjectives are adjectives that occupy predicative positions, that is, come after copula verbs like 'is' and 'make'. If an adjective precedes a noun, it is said to be in attributive position. NOTE: The descriptors 'predicative' and 'attributive' are not absolute. They are used to describe an adjective relative to the adjective's grammatical context.

When 'alone' occurs in your example (1), it is indeed in predicative position. But when it occurs in (2), it is not in predicative position, which is why it is ungrammatical.

'Alone' is restricted only to predicative positions. We might call it a predicative-only adjective (following @Araucaria). 'Alone' is not the only adjective that is restricted to predicative positions. 'Afraid' is also restricted to predicative positions. Interestingly enough, other adjectives are restricted only to attributive position (for example 'main'). Here is an instructive passage:

Most adjectives can freely occur in both the attributive and the predicative positions. However, a small number of adjectives are restricted to one position only. For example, the adjective main ("the main reason") can only occur in the attributive position (predicative: *"the reason is main"). Conversely, the adjective afraid ("the child was afraid") can only occur predicatively (attributive: *"an afraid child").

Many adjectives beginning with the preffix 'a-' are restricted to predicative positions (for example 'afloat', 'awake'), as pointed out by @Rathony. The Wikipedia article on English prefixes even lists the prefix 'a-' as one that turns verbs into predicative adjectives. An interesting question is whether or not all predicative-only adjectives were formed from the 'a-' prefix. As far as I know, this is an open question.

Many adjectives that start with the letter a function only as a predicate adjective, e.g.:

alike, afraid, asleep, alive, awake, alone and aloof, etc.

Some of the above listed adjectives are made with "a + noun". According to Online Etymology Dictionary, aloof comes from

1530s, from a- (1) + Middle English loof "weather gage," also
"windward direction," probably from Dutch loef (Middle Dutch lof) "the
weather side of a ship."

Asleep is from

c. 1200, aslepe, o slæpe, from Old English on slæpe (see sleep). The
parallel form on sleep continued until c. 1550.

The same dictionary explains about a in the above words as follows:

in native (derived from Old English) words, it most commonly
represents Old English an "on" (see a (2)), as in alive, asleep,
abroad, afoot, etc., forming adjectives and adverbs from nouns; but it
also can be Middle English of, as in anew, abreast (1590s); or a
reduced form of Old English past participle prefix ge-, as in aware;