I wonder can we use the adjective after a noun to modify the noun. I found these examples.
So why can't we say "I happy", " you welcome" as completed sentences? Is there a rule when we modify a noun directly with a adjective follow up? Is it related with omitting? Like,"job that is done"
There are circumstances when some adjectives and participles are placed after the nouns they modify—for instance, when the adjective or participle itself has a following complement. You may read about more examples in the Wikipedia article on Postpositive Adjective.
But you have not given us contexts in which this 'postposition' would be called for. None of the examples you give are constructions you would find in ordinary writing or speech, except under the circumstances I mentioned or as accidental collocations within a larger context. They aren't even what you might hear as brief "elliptical" answers to question.
They are instead the sort of thing you might jot in the margins of a book, or use as a title or caption or headline. Haste, or the need to save space, or the desire to use bigger type to attract viewers, sometimes leads writers to use a special style of writing called headlinese, which is usefully summarized at this question.
You should understand how this style of writing works; but it should not be imitated except when you are composing headlines. For the 'body' of your text, use ordinary syntax, which is much easier for readers to understand—and harder for them to misunderstand!
