I think you are mixing up some different concepts here. To help clear it up, I’ll define one thing before the other. First...
Attenuation = Fermentability. In “normal beer fermentations”, fermentability/attenuation has more to do with the specific yeast strain selected, mash time and mash temperature, and amount of simple sugar adjuncts, than anything else.
For example, with respect to yeast selection, Strain X has an average attenuation of 80%. Strain Y has an average of 68%. So, if say the two strains are both English strains that give similar flavors, but you want a fuller, sweeter beer, then you can opt for Strain Y, but if you want a really dry, crisp beer, Strain X will probably be the better way to go.
With mash time, the longer you mash within normal parameters (mash temperature in the 140s or 150s for say 40 to 90 minutes), the higher the fermentability/attenuation. And vice-versa, if you mash for just 20 minutes, your wort will not be nearly as fermentable/attenuable.
Mash temperature, as most of us know, if you mash at 148 F you will generally have a more fermentable wort than if you mash at 158 F. (However, as I understand it, Denny is experimenting currently with the high end, mashing in the 160s and not finding any decrease in fermentability compared to low to mid 150s. So, all rules of thumb can in fact be challenged, and can vary, through experimentation! Too many variables!)
Simple sugars: If you throw 2 pounds of table sugar or candi sugar or brown sugar or honey or corn sugar or molasses or any other kind of simple sucrose/dextrose sugars into 5 gallons of brew, then the average fermentability/attenuation of your 5 gallons of beer will increase since those sugars are nearly 100% fermentable. In fact, when a lot of simple sugars are present (e.g., mead and cider), then the specific gravity will often continue down below 1.000 until it hits some really low number around 0.992-0.994. So the “apparent attenuation” compared to 1.000 is actually GREATER than 100%!
Now... secondly... what you are also asking seems to be about fermentation temperature, cleanup, and ABV.
With respect to fermentation temperature, well, every yeast strain seems to have a certain low temperature limit, below which it really doesn’t perform very well if at all. Strain X might not ferment at all below 62 F, while Strain Y is happy to ferment all the way down to 35 F. Each strain is quite different. However, luckily, with most ale strains, you can go as low as 60 or even 55 F before it poops out... most of the time. Belgian and weizen yeasts are much more finicky and love higher temperatures. They’re all different. So, you kind of have to learn all the little tricks for each one, and luckily, the yeast manufacturer and the interwebs are full of information on almost every commercially available strain so that you don’t need to guess.
And now getting back to those Belgian and weizen strains, yeah, they like warm or even hot temperatures, but the result might not taste as great as if you ferment at the lower end of the spectrum. So what a lot of folks today in the 21st century are doing is pitching these yeasts in the low to mid 60s F, fermenting there for a couple of days, then bringing the temperature up to the 70s or even 80s to finish up. This is more for flavor impacts than for attenuation, however it’s certainly for both reasons. If you did the opposite, let’s say, and you pitched and fermented at 75 F for a week and then decided you wanted the benefits of a cool fermentation and brought things down to 65 F, well, most yeasts and especially the Belgians and weizens will poop out early and just not want to get the job done. So it’s important to always start low and end up high, rather than the reverse, if you want to maximize both attenuation and flavor awesomeness. The two go hand in hand it seems.
Cleanup... this has nothing to do with attenuation, but everything to do with flavor. Sulfur and diacetyl are generated in all fermentations. Sulfur compounds (smell and taste farty, rotten egg, match-heads) can be absorbed by yeast and can also bind up with other things and settle out, and usually do within a few weeks of mid to end of fermentation. So there’s not much special you really need to do there in my experience. On the other hand, diacetyl can be eaten by yeast, but the yeast often needs some encouragement to get the job done – if they don’t eat it, then it stays in your beer for years. So that’s why we like to help out the yeast by raising the fermentation temperature by 5-10 F near the end of fermentation after the beer already tastes like good beer with the possible exception of diacetyl and other little things. The yeast that are beginning to fall asleep then wake up more and say to themselves, okay, what’s left to eat around here, and they see the diacetyl, which is NOT their favorite food, but by raising the temperature, you’ve put them in a hungry mood so they say to themselves, well okay, I guess I’ll eat this crappy diacetyl, as long as there’s still a little maltose to sprinkle on top to make it taste okay. That’s essentially what’s going on. How’s that for a story! I’m sticking to it.
The consumption of non-sugar materials such as diacetyl and sulfur is NOT attenuation. Attenuation is about consumption of sugars to turn them into alcohol and CO2. At higher temperatures, yes, the yeast is happier, and will eat all the sugars faster, so your maximum attenuation will be reached faster. So, does raising the temperature improve attenuation? Well, maybe, but not necessarily. If you bottle your beer, which contains some residual yeast, and you prime it and leave your beer at cellar temperatures for a couple of months, the remaining yeast cells in those bottles are going to ferment whatever they find in there, so your attenuation will be as complete as it’s going to get, even at reduced temperatures in the 50s or 60s. Given time and decent temperatures, the yeast will eventually find everything fermentable and eat all of it until it’s gone. However, if on the other hand you bottle or keg but you cold crash all the beer when there’s carbonation and you leave it down in 30s F all the time until you’ve drank it all, then due to the cold temperatures, below the limit for that strain as we previously discussed, the remaining yeast will most likely never reactivate, so your attenuation could actually suffer, especially if you used any priming sugar or bottled too early without ensuring final gravity had been reached.
Personally, I warm all my beers towards the end of fermentation to ensure final gravity and attenuation have been attained, and then after bottling I leave most of my beer at cellar temperatures until I’m ready to drink it so that the yeast can continue to act and evolve the beer for a few months after bottling. Some brewers do these things, some don’t, and all of us make pretty dang good beer, so, does it really matter?? Well that’s for you to decide.
ABV is related to attenuation because the higher the attenuation, the lower the final specific gravity, and thus the higher the ABV, since ABV can be calculated via the simple equation (OG – FG) x 131 = ABV.
I think the above paragraphs might answer a few of your questions, and somehow I bet they will raise even more questions. And that’s cool. That’s what forums are all about.