It almost seems from the preamble to your question that you are really asking two questions.
- How can true prophetic visions or dreams be discerned from mundane dreams or hallucinations?
- How can I answer skeptics regarding these issues?
I will start with the second of these, your preamble. After many years in the Christian life, and having been a skeptic myself years ago, I have come to realize that God may not be all that interested in convincing skeptics. I do not mean that God hates them (He doesn't), or considers them hopeless cases (He doesn't - I am proof of that, and thankful for it). But skeptics, almost by definition, have their minds and hearts set against the gospel or the reality of God.
Have you ever wondered, or been asked by a skeptic, Why does God not show His reality and His power openly, which would convince the entire world to worship Him alone?
There are multiple examples in the Scriptures to indicate that God reveals Himself primarily, perhaps exclusively, to those who seek Him.
The Pharisees and Sadducees came to Jesus and tested him by asking him
to show them a sign from heaven. He replied, "... A wicked and
adulterous generation looks for a sign, but none will be given it
except the sign of Jonah."
Behold, I stand at the door and knock. If anyone hears My voice and opens the door, I will come in and dine with him, and he with Me.
And there is Paul's diatribe concerning the pagans of Rome in Romans 1:18-32:
... Furthermore, just as they did not think it worthwhile to retain the knowledge of God, so God gave them over to a depraved mind, so that they do what ought not to be done. They have become filled with every kind of wickedness, evil, greed and depravity....
To me these all seem to be reflections of Proverbs 3:34:
Though He [God] scoffs at the scoffers, He gives grace to the humble.
This Proverb is not obscure, it was quoted twice by the Apostles in their letters: James 4:6 and 1 Peter 5:5.
Even Paul's own conversion, due to the Lord's personal appearance to him in a vision, cannot in my opinion be considered an exception to this. As a persecutor, Paul still had a zeal in his heart for the God of Israel. He did not persecute in order to build up his own reputation or political power. Paul was misguided, and Jesus corrected him in spectacular fashion. After that, he described himself in his letters as "a prisoner of Jesus Christ", etc. and became one of those he once persecuted with equal zeal.
Beyond these matters internal to the heart, we have what might be considered a curious gospel passage. In Nazareth, the people knew Jesus and his family, and disbelieved. Matthew 13:58:
And he did not do many miracles there because of their lack of faith.
Mark 6 tells the same story. Why would unbelief prevent the doing of miracles? Perhaps that is the wrong question. Maybe it is not that unbelief nullifies the power of God. Maybe instead, unbelief is an impurity and a profanity of the heart, with which the pure and holy things of God cannot co-exist, being opposite each other. And maybe it is therefore a mercy of God that the pure and holy miracles are not done in that situation, because that would intensify the destruction of men's souls (Lamentations 3:23; Genesis 3:22).
Why would that be true? It occurs to me that when the Pharisees and Sadducees demanded a sign, they were, perhaps ignorantly, tempting God. And a skeptic who demands that God show His power, in order to believe that God even exists, certainly tempts God. But God does not serve us. God is not tempted, or mocked. Meaning, God does not respond to these things, unless with a just wrath. Were God to reward this tempting or mocking, it would empower those who tempt and mock. Just imagine what would follow - people demanding more and more 'proof' from God Almighty, in effect making God their servant, elevating themselves above God. Is that not Satan's ultimate goal? This cannot be!
Therefore, while we as Christian believers are missionaries to the entire world, and while God genuinely and deeply loves every one, and our words and actions should be reflections of that love at all times, still there is a spiritual reality at play when unbelief is involved. God does not, or possibly in a sense cannot, reward unbelief through miraculous things such as visions, healings, or manifestations. Nor, it seems, should we attempt to do so.
In my own personal experience, some of the dreams or visions that I have received, that I now believe to be from the Holy Spirit, have served as infallible proof to me of certain things, of a personal nature. But none have been of such a nature that could be used as evidence to convince a skeptic of anything at all. Incontrovertible proof to me, but just a story to anyone else. I think this is how God prefers to do things. He isn't interested in making a big show. Not until the Day of Judgement, that is. For the time being, He speaks in whispers to those who seek Him. There are healings, but these seem to happen for people who are seeking, not for skeptics (there could always be exceptions I suppose). And anyway skeptics are not generally convinced by such things.
Now to the second part of your question. How can one discern whether a dream, or a waking 'vision' experience, is a genuine message from the Holy Spirit or something else? I agree fully with @Mike's criteria, which I believe are summed up by 1 John 4:1:
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God, because many false prophets have gone out into the world.
By "spirits" John does not mean 'ghosts'. This term can apply to teachings, ideas, feelings, urges, thoughts, dreams, even social trends. Clearly John is advising us to consult our God-given reason, and our understanding of the teachings of God.
I believe we would do well, if we believe we have had a prophetic experience, whether a waking vision or a dream, to consider carefully:
- Does the waking vision or dream contain any verifiable details that were not known to you beforehand? (Note: even if such details are confirmed, this should not necessarily be considered proof of authenticity! Pharaoh's magicians performed wonders.)
- Your own maturity level in your Christian walk. What wisdom can you bring to bear for discerning this?
- Whether you are firmly grounded in your doctrines and understanding. Someone with relatively weak understanding may misinterpret a real vision or dream, or mistake a mundane dream for a divine revelation.
- Certainly if receiving treatment for any mental illness in the neighborhood of schizophrenia or psychosis, this is serious grounds for doubt. Insanity is a state of captivity to demonic influences.
- If you had a fever at the time, this is grounds for doubt, but should not automatically disqualify.
- If you are a drug user (including certain prescriptions!) or alcohol abuser, this is grounds for doubt, but should not automatically disqualify. There are plenty of dramatic and miraculous salvation stories of people in situations like this.
- Depending on the subject matter at hand, do you have any kind of emotional over-investment in this area, that would logically lead to doubt?
- How is your prayer life doing?
- Is there any kind of moral failure in your life that still has a hold over you, in practical terms or because you have not yet repented? If so, this could be reason to doubt the veracity of a possible prophetic vision or dream.
These questions are relevant because you need to cleave as closely to God and His ways as possible in order to hear Him clearly and properly. Some of these questions are difficult to self-evaluate. Consult with an elder in the faith, whom you deeply trust, and who will not himself be skeptical of the concept or reality of prophetic visions and dreams. The question at hand is not whether God speaks through visions and dreams (He does), but whether He has spoken to you in this way.
Also keep in mind, many (not all) prophetic messages are given to multiple people (and sometimes NOT given directly to the intended recipient!) as a means of confirmation. Search for confirmations of this type.
And of course, and this should go without saying, pray and ask God for wisdom. Maybe fast over it.