I would like to use metal trim for the aluminum backsplash behind the stove; the trim would go between the backsplash and the cabinet and tile. What kind of trim would you recommend and what is the best way to install it?
I'd go with Aluminum Flat Bar. Very easy to work with for drilling, scribing, mitering, cutting, beveling & just gluing up to the backsplash whether you want to frame the backsplash or just top it. If you want a to fill an upward & outward gap then the Aluminum Angle or Square Tube options in the link above would fit the bill perfectly.
A metal U-Channel trim, miter the corners. U-Channel trim is easy to find in plastic or rubber but it is possible to find it in metal. You might need to search, but it is out there. Look for metal or welding supply shops in your area.
If that doesn't work, have the edges given a flat hem (basically folding the metal edge back on itself).
If you don't have access to either of those two, cut the metal to size and grind the edges smooth. Skip the any plastic or rubber trim as they will just break pretty quickly in the kitchen.
Install with construction adhesive between it and the wall.