The only way to have clean sweeps is to start very slow and put a lot of care into muting (whatever muting technique you're using). Then slowly go up to speed, always using a metronome. Do not rush it, it is a long process and you can't do it in a day or a week.
If your muting is correct you can sweep cleanly no matter the amount of gain. In fact, I suggest you to practice with more gain than you use normally, so that if your muting is not perfect you'll notice it and correct it.
Using the neck pickup helps (less noise) and is the way to go when you perform, but when you're practicing use both pickups: the bridge pickup will reveal better the imperfection of your technique.