No, it shouldn't affect the time of fermentation, although it will change the fermentability of the wort, at least a bit. A stout mashed at 158 may stop at 1.020, while a cream ale mashed at 147 may stop at 1.006. But they shouldn't really ferment any faster or slower, based on mash temp.
One thing that makes a big difference in the final beer is taking care of the yeast- this may translate to a quicker fermentation, but only indirectly as one of the causes of a sluggish fermentation is underpitching the yeast.
Pitching the correct amount of yeast (most homebrewers underpitch by just buying a "smack pack), controlling fermentation temperatures, and using quality ingredients means a healthy fermentation. If you provide optimum conditions for the fermentation, there wouldn't be many "off flavors" for the yeast to "clean up".
One of the things that happens is that a poorly made beer WILL get a bit better with some time. That's not really due to conditioning, it's due to the flavors getting some time to fade. But a poorly made beer with off-flavors will never turn into a great beer. Some esters may fade a tiny bit, "hot" fusels may mellow, too much roastiness may fade, and so on. But the easy solution is to make the beer correctly with proper techniques and not have to give the beer time to be drinkable. A month in the primary won't make a crap beer a good beer, no matter what.