Which ones are correct:
1-He is weeping in shame that he lost.
2-He is weeping that he lost.
3-He is crying that he lost.
4-He is laughing that he won.
Sentence 1- is rather interesting. The relative clause 'that he lost' appears to be an object complement modifying the noun 'shame':
1- He is weeping in shame that he lost.
=> the shame is that he lost.
But appearances can be deceptive. The clause 'that he lost' expresses a reason, i.e., the reason is that he lost, so subordinating 'because' is best:
1-He is weeping in shame because he lost.
2-He is weeping because he lost.
3-He is crying because he lost.
4-He is laughing because he won. (Also, He is laughing at the fact that he won, wherein at the fact can be omitted: He is laughing that he won. I believe 'at the fact' works here because of the verb 'laugh').
The difference between 'that' and 'because' as subordinators: 'that' can be omitted.
He cried out (that) he lost. (Noun clause; Object of the verb)
He cried because he lost. (Adverbial phrase; gives the reason)
Here's a wonderful source on the differences between that and because: