Because of the short neck of the 357 Sig case, you pretty much have to stick with flat point or hollow point bullets. I've used a swaging die to make round nose bullets into flat points, but that's a lot of extra work, and there are plenty of 9mm bullets that will work in the 357 Sig. Any round nose 9mm bullet will give you fits if you try to use them in the 357 Sig, so just skip them and save yourself some heartache.
You have to watch two things. The first is the overall length of the round, and the second is that there is enough bearing surface on the bullet for the short neck to grasp. You'll find that the Hornady XTP 124 grain bullets will work fine, as will the bulk Remington 124 grain hollow points. Don't try to use the Remington Golden Sabres, as they are a stepped bullet and won't work in this round.
I've got one pistol that prefers 115 grain bullets in 357 Sig, and gives better accuracy with them than the 124 grain bullets. The other one prefers 124's. For practice, I shoot either the Berry's 124 grain 9mm flat point plated bullets or 124 grain plated hollow points. They are both very accurate over 13 grains of AA #9.
As for the variance in bullet weight, that's normal. If you've ever toured a bullet factory and seen how they're made, you would marvel at how consistant they are, especially when you consider they're made by the millions. Most bullets are made on WW II punch presses. Both the Sierra and Nosler bullets are made on those machines and I've toured both plants.
Hope this helps.
Fred