Then, the sphere will float when B>Wtotal, otherwise it will sink. Is this correct? So the air inside the sphere will affect sphere's ability to float.
Yes but it will be a small effect. The density of air is nearly a thousand times less than that of water.
My confusion is that I assume that I have already considered this effect during the aforementioned calculations
The big question becomes how are you measuring your masses, both of the sphere, the objects you place in the sphere and the density of the water you are putting your sphere in?
Your scales are sitting in air, so they don't measure the actual weight of the object they measure the difference between the object's weight and the weight of the air displaced by the object.
So if you weigh everything, water, sphere and objects using normal scales in normal air you are implicitly taking account of the mass of the air by pretending everything (including the water) is slightly less dense than it really is.
I) But haven't masses in vacuum not the same attraction and speed.
No. Their weights are different, so they are not "attracted" / pulled in by gravity equally.
Think of this: If you find 100 heavy perfectly round stones, and you put 5 plastic balls full of air with exactly the same size in the basket with them, what will then happen when you shake them a bit? Will the lighter plastic balls fall to the bottom or "float" to the top?
They will float to the top.
The point simply is that it is easier for helium atoms to move up than for air molecules. If you shake the basket violently, the stones might jump a bit while the plastic balls can jump much higher. So on average, the helium atoms will move much higher upwards, and as soon as they do that, some oxygen molecules will take their previous location. Now they have a new location higher up, and the same happens.
Overall this causes the effect of buoyancy, sometimes called updrift, which is the force that this lighter material is pushed up with. And this upwards force is exactly the same as the force, with which the heavier materials pulls downwards - in other words, the lighter material is pushed up with the weight of the displaced heavier material, which now pushes to come back in place.
This was Archimedes' discovery.
Now to your other sub-questions:
Can be said that for airmolecules the atmosphere is a vacuum?
Well, no, a vacuum is a vacuum. If there are molecules present, it isn't vacuum, and the atmosphere isn't a vacuum.
So all together helium should have the same attraction to earth as the other airmolecules?
No, their "attraction" to Earth are different, because that "attraction" must be weight. And the helium atoms weight is lower.
II) Because helium atoms are much lighter, perhaps they could have a higher speed than fe O2 or N2?
Mass (or weight) doesn't influence possible speed. It only influences how hard it is to make them reach the speed.
Ok, but those helium atoms are in a balloon so they pushes at all sides of the balloon equal so the balloon shouldn't move at all?
If only the balloon with helium was present, and no gravity or outside atmosphere, then you are completely correct. The inside pressure cannot make the balloon move. But with gravity present, the whole thing is pulled downwards, and with the atmosphere present, there is a buoyancy force upwards as discussed above. Which-ever of these forces is greater, makes the balloon move.
III) When a balloon starts ascending from the ground there is more air (pressure) above him than beneath. So the airpressure above him should push the balloon to the ground?
Incorrect. You actually said it yourself just before: Inside the balloon, the pressure equalizes throughout so the push at any point on the balloon is the same. Same goes for this air column: All the air in the column above presses down, but the tiny bit of air below pushes up with the same force to balance out the pressure.
Best Answer
I guess you have mentioned all significant factors. Just a few comments.
Gravity reduction with altitude is not significant for altitudes accessible to balloons.
Pressure is a rather complex function of altitude (http://en.wikipedia.org/wiki/Barometric_formula).
You should take into account the elasticity of the balloon envelope to calculate the balloon volume as a function of altitude.