Many people use the nets, i don't like them exactly because of the reason you read. If you NEED to use a net because there's no other way for you to get your fish out of the tank, get a brine shrimp net. These are VERY soft. The fabric is white and not see-through.
I prefer to scoop my fish in a cup and pour him into the new water. Well, I used to anyway. I no longer do these types of water changes as now I have cycled tanks and I dont remove my fish during water changes.
anyway, this method is very easy, specially if your betta likes you because he will approach the cup on his own and that combined with the suction action of the water will bring him right where you want him.
It's the best way to avoid damage to their scales and fins, IMO.
Two galons if ok for your betta. If you're doing partial water changes, I suggest 50% twice a week. But at some point you need to wash that substrate if you're not using a siphon for the changes.
When I used to follow this, I siphoned my tank every other partial water change. For the other change, I just scopped out the water. The fish never left their tanks!