1) They don't add more damage, you're right. To increase damage done, you need to use the Sphere Grid and activate Strength nodes. The point of having the different weapons is, as you say, to add different abilities. When you get Rikku permanently later on in the game, she adds the Customise function to the main menu, allowing you to use different items to add abilities into free spaces in your weapons/armour (which is why weapons and armour with four spaces in are quite valuable, both in terms of how you can customise them and also in terms of price for when you sell them).
2) If you can find a path into Auron's grid it'll probably be blocked by a "level" sphere node, which means you need to find a level sphere that corresponds to its number (eg. level 2 sphere node needs a level 2 sphere). As far as I can remember, when you have finished Kimahri's small sphere grid, you can choose which path to follow, but I think some are blocked by 2+ nodes when you only have a few level 1 spheres in your inventory at that time. I went down Lulu's path, but that ended up being rather useless, so if you can go for a heavyweight route like Auron's, then give it a shot, as it'll be more useful.
I haven't tried it, but I understand that the "expert" sphere grid is the same layout as the standard one, but each player starts in the center, so you can customise each character right from the beginning, essentially meaning you can choose their class rather than going with their predefined ones (Yuna being a healer, Auron the heavyweight, and so on).
