Can a feeling that God exists count as a good reason for believing in God? Could it also count as good reason for public policy -- for funding churches and religious schools?

Hi, great question! I will focus on your first question, because I think your second question, about public policy, requires discussion of all sorts of things about the distribution of goods in a society, and the law (in the US, for example, we have to keep in mind that our constitution seems to forbid any such funding). Perhaps another panelist can take up those issues. I'll stick to the question whether the feelings that God exists can count as a good reason for believing in God.
First I should point out that many philosophers have written on this, and have different views (shocking! I know). Some, writing in the Calvinist tradition (e.g. Alvin Palntinga and William Alston) think that some feelings that God exists are like perceptions, or sensations, of God, and they should count as evidence for God's existence just like a perception of, say, a table counts as evidence for a table. They both have arguments (different ones) to the effect that one can rationally regard one's feelings about God as perceptions of this kind before one already concludes that there must be a God. Intuitively, this might make some sense. You might think to yourself: hey, I have this feeling, out of nowhere, that there is a God, and since I don't have any evidence to the contrary, why shouldn't I regard this feeling as some indication that there is a God? On the other hand, dig a little deeper and some doubts begin to appear. What best explains the fact that you feel that God exists? Is it a need, or a wish, that you have? Is it the product of, say, your need to feel comforted in certain ways? Is it the product of your desire to think that your upbringing was righteous (if you had a religious upbringing) or all wrong (if you had a secular upbringing)? Is the feeling rooted in a very deep and desperate desire not to die? These suggestions, and many complex variations of them, have all also been offered as alternative explanations of your feeling that God exists. Unless you settle where the feeling comes from, it seems a stretch for you to regard it as a good reason to believe in God, doesn't it? Hmm. Maybe not. Some *other* philosophers argue (erroneously, I happen to think) that information about the causal source of your "evidence," or what you take to be evidence, is irrelevant to whether it is rational for you to believe on its basis.
Ok, so I just said, essentially: lots of people say different things, some regard the feeling as evidence, some don't. So, maybe that doesn't help you. I'll try to help by making two further points. First, there is an ambiguity in the term "reason" in your question, and, second, it may be useful to compare your feeling that God exists to other feelings.
First, what counts as a "good reason" to believe something? In my first paragraph I assumed that only evidence can count. But maybe the feeling you have in mind, perhaps a pervasive, ever-present loving feeling for something greater than us, provides a different sort of reason to believe. Perhaps your feeling makes it a good idea, from a practical perspective, to believe. Some philosophers (most famously Pascal) have thought that practical considerations can make it rational for us to at least try to believe. If believing seems good for you, given your feelings, maybe that's a reason to try to believe it. How you get yourself to believe it is a whole other story (I recommend an essay, basically just a transcription of a lecture, by Daniel Garber, entitled 'What Comes After Pascal's Wager' on this topic). Anyway, my point is that your feelings may provide some non-evidential sort of reason to believe. The downside of thinking this way is that, on reflection, you might conclude that nothing but evidence is a good reason to believe. Believing is taking something to be TRUE, so shouldn't any reason to believe be a reason relevant to the TRUTH? You have to figure that out first.
Second, it might help if we consider feelings about other things, to compare with your feelings that God exist. Suppose I am desperate for money, perhaps I have a very ill relative. Suppose I've run out of other options, and decide that attempting to rob a bank is the only way. This robbery is very risky; I could get shot or arrested, making things even worse for my sick relative. But as I consider it, I am overwhelmed by a feeling that I CAN DO THIS, THIS WILL WORK. Not because I have any evidence that I won't get shot or arrested. It's just a feeling that this is the right course of action and that it will end well. Now, is my feeling a good reason to believe that it will all work for me out when I rob a bank?
Objection! That example is unfair. In this example, I had some reason to think that the bank robbery will not work out. My feeling goes AGAINST my prior knowledge about the robbery plan. But, you might say, your feeling that God exists may occur when I have no prior knowledge, either way, about how likely it is that God exists.
A different example, then: my daughter's birthday is in March. Right now, it is impossible for me to tell how likely it is to be raining in March. But she loves the beach, and I want to plan a grand party for her, now, on the beach. Suppose, for the sake of the example, that I have to reserve the party location now if I want to have the party there in March. Also, suppose I don't know how to use google and so I lack access to average weather in March on this beach. I just really don't know how likely it is, and suppose I lack any good reason to think anything at all about the probability. Some might say: you should think the chance of rain is 50%. Maybe. I myself think that's wrong. But, anyway, I don't have any background knowledge to the effect that it will rain, or that it won't rain. And yet, I have this feeling, when I look at my daughter, that she will LOVE her birthday party on the beach, that it will all work out and there will be no rain. Is this a reason to believe that there will be no rain?
I think that we want to say "No." It's not a good reason to believe that it won't rain. However, I think it is, perhaps, some reason to be hopeful, and to plan for a rainless birthday party. Maybe. At least we get this out of the example: a feeling that it will not rain, or that God exists, may warrant some other attitudes, besides belief, about rain and God. It is worth thinking about that, I think. Let's not pretend that belief is the only thing that matters.
I hope that helps!

Read another response by Yuval Avnur
Read another response about Emotion, Knowledge