Is it wrong to use the phrase "The alive animal"?
Is it alright to say, "The animal was alive."?
The animal was alive is a correct sentence. The alive animal is a noun phrase and therefore not a sentence, since at the very least it is missing a verb.
Having said that, I interpret the question as asking whether it is possible to place the adjective alive in front of a noun. In other words, can alive be used attributively? Most adjectives can be used both attributively (before the noun) and predicatively (following a copula such as the verbs to be, to seem, to look). So happy can be used both attributively and predicatively:
But some adjectives can only be used predicatively; they cannot be used attributively. Alive is one such adjective, so the alive animal is ungrammatical. There are many other non-attributive adjectives beginning with the letter a. For example, it is ungrammatical to say:
Other adjectives can only be used attributively; they cannot be used predicatively. For example: