When a positive point charge is brought near a conductor like a metallic sphere, does the sphere - get attracted to the charge, because induction causes negative charges to move towards the charge and positive charges away from it, and hence create a net force on it? - or should it stay at rest, because electric field and so electric force on any electron inside the metallic sphere is zero and there would be no acceleration at all? I am confused, as both appear to be right.
The former is the case.
The electric field being zero just means that all electric forces balance (and nothing accelerates anymore). It says nothing about the electron-arrangement of any electric charges inside a conductor. You can easily have a non-uniform arrangement, with electrons in the electron-dense part repelling each other while they collectively attract the positive half. Naturally an extra electric force must keep them in place, and this is the external electric field you provided.
This can keep them all in balance, causing the net internal electric field to be zero. The external electric field is non-zero at the same time. The external charge is naturally attracted to the electron-dense part that it is nearby, and vice versa.
In non-conductors you have less freedom of the electrons and other charge-carriers to rearrange and you thus do not necessarily see much charge induction.
The positive point charge produces an electric field around itself.
The metal sphere cannot have an electric field inside it so in some way it must negate the effect of the electric field produced by the point charge.
It does this by redistributing charges with there being a net negative charge on the side that the positive point charge is positioned and a net positive charge on the other side.
These are called induced charges and have the result of making the net electric field inside the conducting sphere zero.
Now you have a situation where the induced negative charges are closer to the positive point charge than the positive induced charges so there is a net force of attraction between the conducting sphere and the positive point charge.
If the positive point charge is allowed to move it will move towards the conducting sphere gaining kinetic energy at the expense of the loss os potential energy of the conducting sphere & positive point charge system.