Gravity vs inertia As stated according to Newton laws of gravity, every object with mass attracts all other object with a force which produces acceleration.  Basically there are several forces in the universe which affects our planet as well.
My question: can an inertial frame (which has net force zero) can exist in these condition, and does this frame plays any role in producing acceleration? 
If two objects are many miles apart and one object has much greater mass then, will the heavier object still accelerate relative to lighter one?
 A: Inertial forces are considered non-assigned forces and thus are regarded as fictitious forces (even though they are real and the observer experiences these forces). 
They are mostly related to relative motion between (non-inertial, accelerated) frames of reference and transformation(s) thereof (eg centrifugal, Coriolis force, etc.. ). 
Apart from that, Newton's gravitation is an inverse square law, which means the gravitational force is diminished by square of distance while being analogous to the masses of objects.
Newtonian Dynamics were based on an absolute space and time. Indeed in Newton's era, there was (and still is) great discussion concerning the centre of the universe which was to be taken as an absolute reference point (for formulating the equations of motion for other systems)
Einstein's special (and general) relativity had an insight on this issue, of absolute reference, whose meaninglessness becomes one of the starting points of SR, GR.
The principle of equivalence, as used by Einstein, in GR (general relativity), actually takes these accelarations (based on non-inertial frames of reference) to be the analog of the graviational field (meaning each such acceleration comes from gravitational field and vice-versa). And this elucidates the equivalence between inertial mass (as used in Newton's 2nd Law of motion) and gravitational mass (as used in Newton;s gravitational law) (which has been experimentally verified up to $10^{-8}$ )
Inertial frame of reference, wikipedia
Absolute space, wikipedia
Newton;s views on space and time, SEP
