Hi Brian,
Balls that are purchase in the Pro Shop gradually wear and expire. Some faster than others and that's represented by the durability rating of the ball. The more bubbles indicated on the chart in the pro shop the longer the life of the ball. Strokes taken with the putter do not decrease the durability. Unless you are hitting them OB or in the water you will never go through 6 balls in one round.
To help put it into perspective, consider that most professional players change their ball in competition every 3 or 4 holes. That's usually because they generate so much force and spin that the grooves on the club gradually wear the ball down or misshape it. The balls you have now (Tour-SD) are some of the lowest durability offered, but even they will generally last a little more than a round. Its lifespan is entirely dependent upon how many strokes you take from off of the green (remember putts don't count). Also, don't forget that every player is provided with an unlimited amount of WGT Starter Balls.
The hits meter shows you how much durability is left on the ball in play. Once the meter decreases to 25% it lets you know that ball is nearing the end of it's lifespan and is becoming worn. You don't have to start with a brand new ball each game. You'll start with the same ball you finished the last round with until that ball expires or is hit off the course. Performance does not change whether there is one hit left or a brand new ball. The hits meter has no impact on your shot, it's only there to show you how much life is left on the ball in play and it is not included in the calculation of your shot.
In competitive play the one-ball-condition prevents players from switching the brand or model of their ball. You must play the same type of ball that you began the round with (ie. If you start with a ProV1x, you cannot switch to a ProV1) and we employ a variation of that rule here. This rule is used on the various PGA Tours and in many USGA Championship events as well.
You can check how many balls are in your bag at any time by clicking on your name in the game, clicking the equipment tab and then clicking the ball that is equipped in the top right hand side of the screen.
If you have any more questions or concerns about the game you are welcome to check out the FAQ. A lot of useful information about the game can be found there and all players are encouraged to read it over before diving into the fray.
Hope that helps =)