Reusing old circuit components
Determining whether they still function is easy, determining their lifetime left is hard to impossible.

In general, visual inspection should tell you whether a component is worth further testing, charring/bulging/discolouration/physical damage is indicative of a dead or abused component and not worth salvaging


Capacitors if your multimeter has a capacitor setting, checking to see if the capacitance is within the tolerance of the capacitor rating. Electrolytic capacitors tend to lose their capacitance over time, and have an effective lifetime of ~15 years. Checking other parameters may be useful too, the leakage current should be very low, and the DC forward resistance should not be measurable on your multimeter. 
Resistors check that the resistance is within the tolerance of the rating, check for abvious signs of past overheating. Resistors tend to either work or not, excessive overheating can change the resistance and can burn out the resistor.
Diodes, with a multimeter you could check that it still conducts current in the right direction and measure the voltage drop across the diode, ideally you would use an IV curve tracer to measure IV characterstics and check against specs, an IV testing setup can be made with a computer controlled DAC and some electronics.
Transistors, check relevant forward voltage drops across spec, stick in IV curve tracer to verify junction to junction properties.
Thermistors , using warm water at different temperatures and a ziplock bag, check that the resistance across the thermistor is within spec for a given temperature. 
Arbitrary IC's, The datasheets all IC's include a plethora of tested characteristics for the IC such for instance, opamp leakage current, time delays in IC logic. At the end of the charts there is usually a diagram of the test setup that can usually be emulated by amateur electrical engineers and hobbyists with only the most essential test equipment (oscilloscope and signal generator primarily). Compare the device of interest to its spec sheet tollerances.


However, in the end, whats the point? Passive components and simple active components like transistors/fet's/and regulators are very cheap to get if bought in quantity, if you are trying to build up your component stock its much easier and cheaper to buy "grab bags" of componenets available from many suppliers, they usually give you many values and varieties of components, often priced around 2-3 cents a part. Higher end components like DSP chips, microcontrollers, interface logic/etc. can usually be obviously determined if its still ok (does the old device still work? or did that part of it work before it broke for a different known reason?). The biggest potential for damage is probably in the desoldering and resoldering.