Sentence 1. "The man denied stealing the camera." is acceptable and often used. It is the equivalent of 'The man denied [ the act of ] stealing the camera. Using the simple gerund as a type of activity/noun shorter. A fuller version of this sentence could be "The man denied [ that he carried out the act of ] stealing the camera." You can see that 'him' stealing the camera is implied but not specified.
Another common way to say this is "The man denied that he stole the camera."
Sentence 2. "The man denied having stolen the camera." This form is sometimes used, especially in writing, but it's harder to justify its correctness. It strikes me as an over elaboration for the sake of formality. Its use may come from another similar use. Certainly "having + past participle" is used for "result participle clauses", when it means "as a result of" or "because" and by implication "after". For example "Having stolen the camera, the man ran away." = "The man ran away after/because/ he stole the camera." or "The man ran away as a result of stealing the camera." In this sense "having stolen the camera" takes on the meaning of, the act of stealing the camera, similar to the gerund "stealing", except it's using the past partciple. So now we have " The man denied the act of stealing the camera [ in the past ]." = "The man denied having stolen the camera."
Sentence 3. "The man denied having been stealing for the third camera." This sentence is not correct. The two parts of the sentence do not fit together in any context that I can imagine. What might be correct depends in what you want to say, what situation you want to describe.
If you remove "for" from the sentence, we get a correct sentence "The man denied having been stealing the third camera.", a form of "The man denied that he was stealing the third camera.", Using "having been stealing" is different to "having stolen" It is used to show continuation, with "when something else happened / while " such as "The man denied having been stealing the third camera, [when she saw / while ] his hand was in her bag."
If the "for" is intended and important in "for the third camera", the sentence must change. For example one context might be that the man admitted that he stole two other cameras but was borrowing the third . We could use "For the third camera, the man denied having been stealing [it]."