Help regarding Rachmaninoff's Prelude in G Major (Does the right hand need to be mathematically precise?)

Hi everyone! I've been trying to learn the Prelude in G Major by Rachmaninoff and have just finished watching the excellent lesson by Boris Giltburg.

I'm just wondering if the melody on the right hand for Bars 2-6 have to come in mathematically precisely with the left hand?

Example in Bar 2, does the F# on the right hand have to come in between the B and G on the left hand, the E and E on both hands have to come in together and the D on the right hand comes in between the B and G on the left hand?

I'm thinking perhaps in Bar 2, the E on the right and left hand must come in together, whereas the F# and D on the right hand have more freedom to come in when they want. 

I'm also wondering if the E on both the treble and bass clef have to come in at precisely the same time? I'm noticing that when I play, sometimes they don't come in together. 

Same thing as the G for the right hand and E for the left hand in bar 3 and the B for the right hand and D for the bass clef in bar 4. Sometimes when I play, I don't play them at precisely the same time. Should I strive to play them at the same time? 

At times, I notice my left hand seems to be playing in a drunken manner? I don't know how else to describe it. Should I try to make sure the rhythm in my left is as strict as possible at all times?

I find I play better when I'm not overly concerned about making sure both hands play mathematically precisely, but rather just allowing the music to just flow. 

I need help and advice on this.