What influences capacitor maximum current handling?
Large physical size will help dissipate power, but a large capacitor may also have large losses due to high ESR. Power supply capacitors sometimes have inverted ribs to increase the surface area and help them dissipate power (though I have not seen that in quite a while, Philips parts used to have them). Low-end capacitors are often physically smaller than those from reputable manufacturers for the same capacitance- avoid those where possible. 

If you're scavenging stuff, you can try to look up the part numbers, or if it's no-name Asian stuff you might try to guess from the circuit function, but if the requirements are demanding, you might be better off to just go buy some parts of known specifications for a dollar or two. Generally even no-name parts will have the part capacitance, the working voltage, and a rated temperature marked on them. 

A general rule is that the lifetime of an electrolytic capacitor will be at least 2,000 hours at the rated temperature (that's about a work-year). Some are rated for more life at the rated temperature. The temperature will almost always be marked on the capacitor- often 85°C or 105°C, but sometimes more. 

The life doubles roughly for every 10°C you can reduce the temperature- internal heating via ESR losses contributes, but also external heating from other components and from the environment. If you want the thing to last a long time, use a high quality part and run it as cool as possible. If you want it to die fast, use a cheap part and run it hot.