Officers are an important target for snipers because they are in charge of not only leading soldiers, but also communicating with higher officers to coordinate the fighting. If you kill an officer, you stand a good chance of disrupting the enemy's chain of command and their ability to fight effectively. If you kill a junior enlisted solider on the other hand, you've only killed a junior enlisted soldier.

Also, there are only so many shots a sniper can take from one position to avoid being pinned down by the enemy. Thus, they tend to focus on the most important targets instead of just killing junior enlisted.

I can't say I got any of this from WWII Soviet sources, but rather from a memoir by John Lewis Barkley, an American who was in an "intelligence unit," and part of his job was sniping enemy officers. He actually tells a story of a CO fussing at him for killing too many German enlisted soldiers for the reasons listed above. It's called "No Hard Feelings!" but was republished recently as "Scarlet Fields."

Edit: For clarification, Barkley also fought in WWI.

Yes, yes, yes&no.

The tactic of targeting the enemy's leaders is as old as war itself. It is common sense that an army with a commander is more effective than one without. As such, you can't isolate it as something the soviets only did.

The idea behind the tactic is to lessen the effectiveness of the command structure as u/onionheadhat pointed out. But more or less the snipers were active on a company or battalion level. Division or brigade commanders were usually deeper behind the front lines and so were not in much danger from the snipers.

This brings me to your third question. Yes, it must have affected the outcome of some battles. Battles are affected by action on the company level. A huge global conflict is not.