Instead of asking how much theology must a believer know lets look at some Scriptures and see what we can find out concerning your questions.
In Gen. 1:1 God simply declares that He is God. He doesn’t explain what that is nor seeks to describe Himself in any way. He simply says God. Why? I think we can find the answer in Rom. 1.
For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness; Because that which may be known of God is manifest in them; for God hath shewed it unto them. For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse: Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. Professing themselves to be wise, they became fools, And changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things.
(Rom 1:18-23)
I know this may sound strange at first, but hear me out . . . You perspective upon the Arminian issue seems to be too God focused. It seems strange to even write such a thing.
The excerpt of Rom 1 that you provided speaks of God and man. But it is not God focused. It is man focused. It speaks more to the failings of men than it does about God. But I believe it is an excellent place to look for the answer to this debate. The text states,
Rom 1:18-23, For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness . . . because, although they knew God, they did not glorify Him as God, nor were thankful, but
became futile in their thoughts, and
their foolish hearts were darkened.
Professing to be wise, they became fools, and changed the glory of the incorruptible God into an image made like corruptible man — and birds and four-footed animals and creeping things. NKJV
The Arminian problem comes more from an exalted view of man than from a low view of God. The problem is that their high view of man is not only incompatible with the biblical view of man, it hampers their understanding of the salvation process. All of this, if brought to its logical conclusion, reflects a lower view of God. But this is where humans are extremely adept. We can allow inconsistencies in our thinking, if we don't think things through too carefully. It doesn’t take very long on CF to see that this is particularly true of people’s theology! From outside the Arminian camp we can see that their high view of man, and the resulting misunderstanding they have of the salvation process has implications regarding their view of God.
You have drawn some reasonable inferences from the stated beliefs of Arminians to determine their view of God, and to assert what it is that they truly worship. Some of those inferences might be inescapable, IF human beings were locked in to being logical, to determine the implications some of their beliefs have on related matters.
However, humans are NOT so infallibly logical. Therefore, it is inappropriate to say you believe ‘x’ and you believe ‘y’, therefore you must believe ‘z’ - EVEN IF the conclusion that if x is true and y is true z must be true. We must allow for the ability of men to be inconsistent. If they assert they believe x and y, but hold z as untrue, we ought to take them at their word.
And now I believe I have committed sufficient time to the defense of Arminian thought.