Magnetic stirrers are just one of many ways to stir up a mixture. If you have two things together that are not completely mixed (that is, one completely dissolved in the other) a mixer simply increases the contact between the surface areas of the two things so that the rate of dissolving can be faster. But once the solution is formed, a mixer has no further function for that purpose.
In your case, trying to add a small quantity of Acetone to toluene or xylene probably won't really benefit from a mixer, whether magnetic bead or not. For the proportions you're talking about, the limiting factor is usually the equilibrium condition of miscibility. That is, at equilibrium (after a very long time, if that's what it takes), what's the limit on the proportion of acetone to toluene in the resulting solution, before you cannot get any more acetone to mix into the solution. There may not be a limit - that is, you might be able to make a solution of these two chemicals in each other at any proportion you like - but if there is, no amount of mixing will change that. And that still is NOT a chemical reaction, so the mixing is not promoting a reaction at all. (Well, technically chemists understand that the formation of a solution really does involve a type of "reaction", but not in the usual sense.) But suppose the limit is "X"% acetone dissolved in toluene. You can make any solution you like with an acetone content of "X" % or less and it will require very little mixing, so a mixer probably is not necessary. But you'll never get over "X" % no matter how much mixing you provide.
Regarding the reference to contamination of your solution from an aluminum container, there are two ways this can happen, and both produce a true solution of aluminum metal atoms (or perhaps aluminum (+++) ions in a soluble complex). As solvents in a true solution, they cannot be filtered out, so your "fuel filter", whatever that is, will note remove them. Since acetone is highly polar it does associate with water quite well. So, adding acetone to a non-polar material like toluene may be a way of adding water to the solution, too, because it can be carried in by the acetone. With this small quantity of water in the solution, even though it starts out associated with acetone, it may react with aluminum atoms at the metal surface to create hydrated aluminum atoms or, more likely, hydrated aluminum hydroxide in the solution at very low concentrations. Another possibility - I really don't know whether this happens to any significant extent - is that, as a highly polar molecule, acetone itself may form very small quantities of stabilizing complexes with aluminum atoms, thus bringing them into the solution.
Soot in the tailpipe is not a result of dirt accumulated in the engine and then later cleaned out. It is a result of incomplete combustion in the cylinder, leaving various unburned hydrocarbons in the exhaust gases, plus mixed-in carbon particles. All of these can deposit it cooler places as black solid particles. It's really the exact same process - incomplete combustion - that deposits junk in the engine (including in the cylinder) and in the exhaust pipe system.
If an engine runs slightly cooler with additives mixed into the gasoline, it is not because they reduced friction in the moving engine parts. It is because the multiple complex combustion reactions have been modified, probably in the direction of reducing the efficiency of the process, thereby producing less heat. After all, thermodynamics shows us that the engine work output is, by the Carnot Cycle, related directly to the maximum and minimum temperatures and pressures produced in the cycle.