The thing is, if God is who he says he is, and I mean the Christian understanding of God specifically, then he is infinitely Good, infinitely Just, and infinitely Loving. He is also the only necessary Being, upon which all of our beings are contingent. He is the prime mover, the first cause, the efficient cause. In him and through him all things have their being.
So, if this God does exist, that seems to trump all of the supposed evidence against him. It is perfectly logical, accepting the premise that God exists, to take the position that when we perceive evil, that there is a good reason for its existence, that, at the very least, God in his infinite nature allows it for a greater good. Put another way, we are finite beings and don't have the big picture (or sometimes even the right perception of the small picture); we just don't have the capacity to take it all in and understand how it truly is and how it is, all together, good. In short, we simply need to remind ourselves that we are not God.
But wait! I'm putting the chicken before the egg, some might say. To take this position, you must accept this understanding of God, but it's precisely this purported evidence against him that inclines me to not accept that. In a way, that's true. But it is still an argument against that doubt, assuming that you are otherwise inclined to accept it. Because if such a God does exist, then this is true, and the problem lies with your judgment, not with God.
And in fact, we can even dial things back a bit, and just go by the purely natural, rational arguments for the Prime Mover, the Necessary Being, the Efficient Cause. You can consider the Ontological Argument, as well. Those arguments, as well, are not without objections, but nor are the objections themselves without objections. Such is the nature of reasoning (and why, again, we should be more inclined to doubt ourselves than God). In any case, it may be not as hard to accept such a God on a purely rational basis, but even so that Being is still the Being upon which yours is contingent. So again, putting yourself in the judge's seat is, well, a bit presumptuous to say the least, and our objections in such case are, at best, irrelevant.
Now I'm not saying that based on this line of thought that suddenly everything is okay and nobody has any doubts about God anymore. What fun would that be? If God wanted to remove all doubts about him, he could; obviously, that's not his thing. I'm just offering this as food for thought the next time you're tempted to put God on trial. How about starting with something for which it would be a little more believable to be in error, namely your own perception or thought processes, rather than the omnipotent, omniscient He Who Is?