If V is e.m.f. of battery and C is capacity of the condenser, then charge given to condenser Q = CV.
As work done = charge x potential
W = Q x V = CV x V = CV²
This is the energy stored in the capacitor \(U={1\over 2}CV^2\)
Energy lost in the form of heat
\(U'=W-U=CV^2={1\over2}CV^2={1\over 2}CV^2\)
\({U'\over W}={{1\over 2}CV^2\over CV^2}={1\over 2}\) which was to be proved.

(i) Electric field at x = 0
\(E={1\over 4\pi \epsilon_o}[{q\over 1^2}+{q\over 2^2}+{q\over 4^2}+....]\)
\(={q\over 4\pi\epsilon_o}[{1\over 1-1/4}]={q\over 3\pi\epsilon_o}\)
(ii) If consecutive charges have opposite signs, then at x = 0, 
Electric field, \(E'={q\over 4\pi\epsilon_o}[{1\over 1^2}-{1\over 4}+{1\over 16}-{1\over 64}+....]^\times\)
\(E'={q\over 4\pi\epsilon_o}[{1\over 1-(-1/4)}]={q\over 5\pi\epsilon_o}\) 

Yes, two balls having same kind of charge can attract each other if charge possessed by one ball is very large as compared to that on the other ball because when two charged balls are placed near each other, they induced opposite charges on the faces of each other. On the ball having small amount of charge, very large amount of induced charge is produced, two balls get attracted towards each other.

It happens because there 9 is an electric field over A but no electric field over B as it lies inside a hollow conductor. Third law of motion is not violated because force arises between A and the hollow sphere, the charge B is simply an internal part of the sphere.

Experimentally the Coulomb's force law is found to remain valid for a separation between the charges of the order of \({ 10 }^{ -15 }m\) to macroscopic distances of several kilometers. It is believed that the law should hold good, however the large seperation between the charges may be.