When I was first delivering FX stems to one of our mixers, I would go through painstaking effort to premix. I would pan and automate everything to be able to deliver a final, re-recording ready stem. It was only after about the 4th or 5th show I delivered this way that I found out that he was deleting everything the moment he imported the session data. Meanwhile he was wondering what the hell was taking me so long... As a result, pretty much everyone here is delivering elements with zero premixing.
So first and foremost, communication up and down the chain is key. If you can make sure that whomever is providing your elements has a clear understanding of what you're expecting from them, then you're not wasting their time or yours.
Now, if you don't have that luxury, or (as your phrasing suggests) the ones providing your elements are just doing a crap job about premixing, here's what I do. Since I've wasted a bunch of time myself soloing up individual elements of a build to see what's what, I start by throwing up all the faders to get a balance. If something sounds out of whack, rather than soloing each channel, I mute where I assume the problem is. Usually I'm looking at meters and waveforms to make my best guess. I've found it faster to listen in the context of the entire sound, mute the offenders to find them, pull them back, unmute and gradually add them back until I like what I hear. This way I'm mixing more of the sound at once, which usually results in fewer passes of the sound.
Now, because I know the guys who are effects building for me, and I know how they work, I know that elements higher up the track constitute more "important" elements to the design. As a result, an effect placed on the "FX 4" track will typically play louder than something on "FX 12." But that's usually just a starting point. So I'll write the basic balance automation, then dive into trim to detail its mix. After I've gotten the effect mixed, I'll use a VCA to feather it in against the rest of the mix.
As far as muting clips goes, I rarely do it. There are occasions where I don't agree with what they delivered and I may have to go hunting for the sound that I think should be in there instead. But more often than not, rather than shift gears completely, I'll EQ or mix down the bad elements sooner than getting rid of them completely. There usually some reason that it's been added to the design.