दिखाई and सुनाई are nouns that mean something like “show / appearance” and “hearing / audition”, respectively. Their exact translation is not important, however. They are used in a very common idiom which has the following form:
X को Y [दिखाई / सुनाई] [देना / पड़ना]
The meaning of the expression is “for Y to be visible/audible to X”. It’s literal meaning is something like “for Y to give an [appearance / audition] to X”. It has a passive sense, although grammatically it is active. It is somewhat like an indirect verb construction with a conjunct verb.
Both दिखाई and सुनाई are invariable; they do not change their form. The verb (देना / पड़ना) inflects to match its subject (Y).
This idiom is more commonly used with देना than पड़ना. पड़ना can imply a transition, namely, that something becomes visible or audible.
This idiom is not used with ने.
Consider the following examples:
आपका चेहरा दिखाई नहीं दे रहा है – “I can’t see your face”
The agent (X को) is often implied. When implied, it usually refers to the speaker. Thus, in the previous example, the implication is “मुझे आपका चेहरा दिखाई नहीं दे रहा है“.
क्या आपको मेरी आवाज़ सुनाई दे रही है – “Can you hear my voice?”
बादल छट जाने के बाद चांद दिखाई पड़ा – “After the clouds parted, the moon became visible”
मुझे आवाज़ सुनाई दी – “I heard a noise”
Note how this idiom is not used with ने, as in the previous example.