I'm by no means an expert in this, but based off of what I've studied, I would say that there is grounds for there being a "soul", but I don't think that there is much for there being God or a god.
In regard to the soul, when I say that I do get the sense that the Buddha believed in a soul, I'm not referring to it in its colloquial sense. In many sutras (e.g. The Heart Sutra), the Buddha says that there is no soul in the sense that every individual doesn't have their own soul that would go somewhere after death or is separate from the body. However, he did believe in something of that sort.
For example, in The Lotus Sutra, in the "life span of the thus come one" chapter, the Buddha addresses the great assembly and says:
"Since I attained buddhahood
the number of kalpas that have passed
is an immeasurable hundreds, thousands, ten thousands,
millions, trillions, asamkhyas.
Constantly I have preached the Law, teaching, converting
countless millions of living beings,
causing them to enter the buddha way,
all this for immeasurable kalpas.
In order to save living beings,
as an expedient means I appear to enter nirvana
but in truth I do not pass into extinction.
I am always here, preaching the Law.
I am always here,
but through my transcendental powers
I make it so that living beings in their befuddlement
do not see me even when close by.
When the multitude sees that I have passed into extinction,
far and wide they offer alms to my relics.
All harbor thoughts of yearning
and in their minds thirst to gaze at me.
When living beings have become truly faithful,
honest and upright, gentle in intent,
single-mindedly desiring to see the Buddha,
not hesitating even if it costs them their lives,
then I and the assembly of monks
appear together on Holy Eagle Peak.
At that time I tell the living beings
that I am always here, never entering extinction,
but that because of the power of expedient means
at times I appear to be extinct, at other times not,
and that if there are living beings in other lands
who are reverent and sincere in their wish to believe,
then among them too
I will preach the unsurpassed Law.
But you have not heard of this,
so you suppose that I enter extinction.
When I look at living beings
I see them drowned in a sea of suffering;
therefore I do not show myself,
causing them to thirst for me.
Then when their minds are filled with yearning,
at last I appear and preach the Law for them.
Such are my transcendental powers.
For asamkhya kalpas
constantly I have dwelled on Holy Eagle Peak
and in various other places.
When living beings witness the end of a kalpa
and all is consumed in a great fire,
this, my land, remains safe and tranquil,
constantly filled with heavenly and human beings.
The halls and pavilions in its gardens and groves
are adorned with various kinds of gems.
Jeweled trees abound in flowers and fruit
where living beings enjoy themselves at ease.
The gods strike heavenly drums,
constantly making many kinds of music.
Mandarava blossoms rain down,
scattering over the Buddhaand the great assembly.
My pure land is not destroyed,
yet the multitude sees it as consumed in fire,
with anxiety, fear, and other sufferings
filling it everywhere.
These living beings with their various offenses,
through causes arising from their evil actions,
spend asamkhya kalpas
without hearing the name of the three treasures.
But those who practice meritorious ways,
who are gentle, peaceful, honest, and upright,
all of them will see me
here in person, preaching the Law.
At times for this multitude
I describe the Buddha’s life span as immeasurable,
and to those who see the Buddha only after a long time
I explain how difficult it is to meet a buddha.
Such is the power of my wisdom
that its sagacious beams shine without measure.
This life span of countless kalpas
I gained as the result of lengthy practice.
You who are possessed of wisdom,
entertain no doubts on this point!
Cast them off, end them forever,
for the Buddha’s words are true, not false.
He is like a skilled physician
who uses an expedient means to cure his deranged sons.
Though in fact alive, he gives out word he is dead,
yet no one can say he speaks falsely.
I am the father of this world,
saving those who suffer and are afflicted.
Because of the befuddlement of ordinary people,
though I live, I give out word I have entered extinction.
For if they see me constantly,
arrogance and selfishness arise in their minds.
Abandoning restraint, they give themselves up to the five desires
and fall into the evil paths of existence.
Always I am aware of which living beings
practice the way, and which do not,
and in response to their need for salvation
I preach various doctrines for them.
At all times I think to myself:
How can I cause living beings
to gain entry into the unsurpassed way
and quickly acquire the body of a buddha?"
I know it's a bit of a lengthy excerpt, but I think it outlines you point quite well. The Buddha said that his life span is countless kalpas, but that through his transcendental powers, he appears dead. One however, can not say that he is living in the normal sense- you can't meet him in person or anything like that. Basically, he is saying that his "soul", is immortal. The term that is used to describe this in my experience is, "life force". Although his body has long been gone, his life force is never born and nor does it die. We all have this, "life force", and for that reason, one can say that he is "immortal".
This is really the closest thing I can think of that really describes what you're terming a "soul".
As for the existence of God or a god(s), I think that there is little evidence for this. Although there is mentioning of gods that "strike heavenly drums" in the excerpt above, that is a metaphor. When the Buddha describes his "pure land", he is referring to the Saha world, saying that, if one acts is gentle, peaceful, upright, etc., one will find that the Saha world is in fact, the pure land.
Another reason that I don't think that there are any gods in Buddhism that the Buddha believed in is just by logic. Some of this is my own personal inference and from my studying of Absurdism and Existentialism, but there, to me, is no reason that there would be a god or God. That however, is personal opinion, so to back it up, here is an exerpt, also from The Lotus Sutra, in the "expedient means" chapter. Here, the Buddha is also addressing the great assembly and, among other things, is talking about the nature of reality:
"What the buddhas have achieved is the rarest and most difficult-to-understand Law. The true aspect of all phenomena can only be understood and shared between buddhas. This reality consists of the appearance, nature, entity, power, influence, internal cause, relation, latent effect, manifest effect, and their consistency from beginning to end.”
This would indicate that reality consists only of things occurring. This reality is then divided into 10 parts as listed above. Suppose there were no living things in the universe. If the sun were to explode and planets to collide, would it really make any difference? Because there is nothing to, in essence, care, it doesn't really matter if there is total destruction of planets or stars or anything. This is the reality we live in. However, we cannot say that no living things exist, for we see them every day. The difference is that, now, were planets to collide and stars explode, people would care- it would then affect living things. Really, when we say something is bad, evil, etc., really what is happening is that the events that occurred that cause our displeasure are just happening to be against what would allow us to live most easily. People say they don't understand how evil or other such things exist in the world. In reality, it exists in our mind. That's not to say that it's nor real, but that it has no existence in reality.
As such, I don't think that the Buddha was trying to manipulate people or cause them to slowly believe in something (at least not God). I think that many people want to believe in God , and so it makes sense for it to exist, but since the Buddha knew that no event has any intrinsic implications, there was really no need for a desire for a god or God. This is the main reason that I see no evidence for a god and nor does it seem like something that would exist.
I hope that helps. If you have any other questions or things that I should clarify, feel free to comment.