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Your Q1 and Q3 are linked together. If you knew the answer to the Q1, you would not have asked the Q3. The answer to Q3 is that the charge density is not constant over the volume, hence you can't factor the ρv out and just multiply by total volume. Instead, you have to do the full and proper integral out, which requires the density distribution function. The answer to Q1 is that the exp(-C ρ z) is a key part of the charge density distribution function. The density varies over the radius ρ and over the distance z.
Contrary to what you say in Q1, charge density is not in general constant over space or time. There are special cases where this might be true enough, but generally it is not true, and in this case, the problem is telling you that it is not true and explicitly tells you the distribution.
For Q2, the extra ρ is a normal part of doing volume integrals in cylindrical coordinates. You probably just forgot that fact. When you integrate over a volume, you are adding the contributions of small little boxes. In rectangular coordinates the box lengths are dx, dy and dz, all which have proper dimensions of length. But in cylindrical coordinates, you can't make a box out of dρ, dz and dθ because dθ does not have dimensions of length. However ρ dθ does have the proper characteristics to form the third length for making a box in cylindrical coordinates.
Contrary to what you say in Q1, charge density is not in general constant over space or time. There are special cases where this might be true enough, but generally it is not true, and in this case, the problem is telling you that it is not true and explicitly tells you the distribution.
What was really confusing was this source material which almost always assume uniform charge distribution in derivations and which I have used in the past. But as you say this text, which I used for my queries above, doesn't assume uniform charge density. Thank you.
One of the multiple choice questions was that if we have two identical charges are placed at x-axis, say Q1 is located at (4,0) and Q2 is at (8,0), then at what place the electric field is zero. My answer was that the field is zero halfway between the given charges, at (6,0). Was my answer correct? Please let me know. Thanks.
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