Learn English – which is correct “to be as flexible as possible” or just “as flexible as possible”

If Longman says "Do not use object to mean 'the thing you are working towards and hope to achieve" and provided "My object is to learn English" as an example, then it is contradicting itself. As for the object-objective distinction, I can only offer an opinion:

1. Objective can be used in any situation where object can be used, i.e. "what's the objet of the game" → "what's the objective of the game"
2. Object can only be used to describe the main goal of something. For this reason, you can say "the object of my visit is to return your book" or "the object of the game is to pick up the most sticks" but not "my object is to learn English" because in the first two, you are describing the central goal of the visit or game, while in the third, you are only describing one of your many goals.

"Don't you tired?" is grammatically incorrect in Standard English, but may be part of the local dialect.

Auxiliary forms such as do, does, did are used with a main verb to frame a question. For example: "Do you speak German?", "Don't you speak German?", where the infinitive speak is present.

But they have no grammatical function in a sentence such as in the one you quoted, because there's no main verb. For example, in the sentence "Don't you tired?", tired is not a verb, but an adjective. Such a sentence is unacceptable in Standard English.

(Note: when we conduct a Google search, it brings up all that it can find in the cyberspace, which is a free-for-all. Since it's natural for us to google for answers, it's recommended that you do it by typing, as in this case, say, "How to form questions in English". You might end up on an authentic site that you could learn from.)