You forgot mentioning Shizuha, Minoriko and Hina, by the way.
Hina actually doesn't count, I think. She leads the curse doll army, so she could just be a doll herself, and she is out-right listed on the wiki as not goddess. Its up in the air, but more likely to fall in the not. I just use goddess for her in CDO because it is convenient and overused in MoF, before anyone calls that up.
As far as Theism goes...
Easily, Faith is not believing that a god exists. You can believe a god exists without giving it faith (Or else it wouldn't be a sought for commodity), But it seems to be needed to be given willingly. Therefore, the transfer of faith is probably willingly done via prayer and other methods. This is why Sanae needs to go out to bring faith in (I'd imagine shrines can act as focus points for faith, perhaps even purifiers) and Reimu can collect faith with just the shrine. Faith is probably a secretion or energy that humans and youkai emit.
Does Reimu herself give faith to gods is a good question. Frankly, she has to upkeep the shrine that is holding the entirety of gensokyo together. I don't think she gives much faith, if at all, to other gods. Much like how Sanae probably gives her faith strictly to Kanako and Suwako. What does interest me on Reimu is IF she has a goddess or god to dedicate faith to. The Hakurei Shrine is connected to the barrier, so obviously the barrier takes faith from the shrine. With this in mind, I do believe its possible there IS no Hakurei Shrine god, and Reimu just isn't sure if there is or not (Say what you want on MoF, I'll note its just her wondering what her own damn shrine god is like.). The Hakurei Shrine seems more based on the barrier than an actual god, so if anything, the barrier is probably the 'god' of the shrine. It is fueled by the faith of the Hakurei Shrine, controlled by the Hakurei Shrine, and is based off the Hakurei Shrine.
Obviously, though, Reimu herself believes that other god/goddesses exist, and in fact may encourage belief in some of them. Her shrine just isn't based on one.
As far as Monotheism goes? Possible as all hell. I note that apparently Faith makes a lot of things go around, and people have even brought up Youkai/God similarities like they -matter-. This leads to the possible belief that the two are just the same thing, and gods just get better treatment. So you could believe in one 'true' god that doesn't require things like faith.
You could also take a scientific method to it, and refer to Godesses and Gods as 'Faith-based beings' or 'Faith-based phenomenons'. The claim would be they're things made of faith that happen to have a shitload of power. They require Faith to sustain themselves like food, therefore they are constantly degrading imperfect beings. Thus, not actual 'Gods' if you were to take the 'GOD IS PERFECT' route, but very powerful beings that are called gods for lack of a better term, or belief they are the closest things to gods. They are however, in reality, a naturally occurring phenomenon, nothing more. This could even lead to a field of study on the manner of faith and faith-based beings, but now I'm just rambling.
Ninja: You could just have a dick god claiming all other gods are posers. Religion does tend to exist on complete lack of some senses.