On voicing, NO, ABSOLUTELY INCORRECT!! Never play the voiced note of a chord even a microsecond earlier or the softer notes earlier. They must be sounded simultaneoulsy yet distinguished from one another. When we first start playing piano, we come to view chords as skyscrapers within the notation. So we view chords vertically. The more we learn about music and voice leading in particular, we come to understand that chords are really not vertical at all. Very often it is more useful to view them horizontally. A good example is when the top notes of the chords (or octaves for that matter) are actually melodic. What needs to happen is that the 4th and 5ths fingers of the RH have to "voice" those notes to give them prominence, in turn bringing clarity to the melody. Thus with clarity, the chord no longer sounds like a uniform cluster of tones all at the same volume, leaving no sense of of what's important and what is not. What is required is that the voiced note becomes foreground and the other harmonic notes of the chord just background. The foreground must always prevail through clarity of playing. In a lesser number of instances, the bottoms are emphasized, primarily using the thumb of the RH. And, with Rachmaninoff, of course, there are countless instances of "lines" running through the middle of sequential chords. Middle voices are always the most difficult to execute. With octaves or chords sounding together in both hands, it's not uncommon to voice the tops in the RH and the bottoms in the LH, although there are exceptions.
How to do it? Take the RH chords that need to be voiced, and practice with that hand alone, undistracted by the LH. Play the melodic top notes (assuming top note voicing) alone first as a connected line to get the feel of it. Add the rest of the chord(s). Now try to attain the sound of the line as isolated by orienting and weighting the hand slightly toward those notes, playing them firmly and with more emphasis with the 4th and/or 5th fingers, depending on logical fingering in the passage. Because you have subtly re-choreographed your hand to accomplish this, the other notes will be deemphasized of their own accord. Repeat many times. Don't be discouraged by apparent lack of progress. Voicing skill is not attained overnight; it must be developed. It might take you two months to even start to become proficient at voicing, requiring endless patience as you work at it. Once you do get it, you'll play with a new clarity, and your playing will be forever changed.
Voicing is extremely important to master, not just for melody embedded in chords. The more difficult situation arises whereby you have not chords, but melody and accompaniment consisting of figuration, both in the RH. The same technique applies. You must voice the melody while keeping the accompaniment quietly in the background at all times.
On your second question: Keep the LH (generally harmonic background) softer than the RH (melodic foreground) by proper BALANCING of the hands. That means simply learning to play the LH mf if the RH is f or the LH pp if the RH is p, etc. It's basically a matter of coordination and dynamic control that comes with careful practice using a keen ear. If you are currently playing both hands at equal volume, you have another major problem to work on there.
Never, under any circumstances, play the LH before the RH or vice versa. That mannerism became popular during the Romantic age in the mid and late 1800s by pianists like Hans von Bulow. Today it is universally seen as an affectation in playing to be avoided as a taboo. Instead, maintain proper balance between the hands, be proficient at voicing, and always play the hands TOGETHER in the process.
I hope this helps you.
Topic: Voicing chords technique