Consider these sentences for example.
Did anyone check they're working?
Did anyone check that they're working?
Did anyone check if they're working?
What conjunction is right to use here or no conjunctions can be used? And why?
Personally I would use whether.
In the sentence "Did you check that they're working?", "they're working" is a condition to be verified. Thus implying "working" to be the desired state.
In the sentence "Did you check if/whether they're working?", "they're working" is a condition whose state is to be ascertained. Thus implying no particular desired state.
"Did you check they're working?" simply implies "Did you check that they're working?"
In cases of true/false or yes/no questions (your case) if and whether may be used interchangeably. In other cases you shouldn't use if.
All three examples make sense as shown. However, "if they're working" carries a lower degree of certainty that they are (as would whether).
"Check that" conveys the confirmation of a presumed fact or condition. "Check if" contains no such presumption.
If lack of presumption is explicit in the sentence, "if" must be used. Otherwise, either conjunction may be used without loss of grammatical correctness.
I checked that he was awake. (presumes a given condition)
I checked if he was awake. (may or may not include presumption)
I checked if he was awake yet. (lack of presumption is indicated by "yet")
I checked that he was awake yet. (grammatically incorrect)
When the object of "check" is a standalone sentence, using no conjunction will seem clumsy to native English listeners and may slightly obscure the sentence's meaning (e.g., the OP's first example sentence in a spoken context).
All of the sentences are valid. However, there is a difference in the words "that" and "if".
"That" refers to a specific thing while "if" refers to a condition. So "that" is more natural to use for something certain (e.g. a statement) while "if" is more natural to use for something uncertain (e.g. a question).
An alternative is to use whether: "Did anyone check whether they're working or not?"