Shottky diodes are literally half a normal diode. They are a metalization layer put on a doped semiconductor region in a way that causes a similar forward/reverse diode operation. The advantage is very low forward drop and very low capacitance and stored charge, so they are also very fast at turning off. They were first popular for switching power supplies where their low forward drop and fast turn off ate up less wasted power.
Leakage is another issue. In a perfect capacitor or diode, there is absolutely zero reverse current flow. It's just like a faucet - leakage is stuff dripping through where it's not theoretically supposed to. And like a faucet, leakage is primarily a waste. A certain amount of leakage can be tolerated, just as you can usually wait a few days to get a new faucet washer. But low leakage is good (with the notable exception of the Millenium bypass!
) in almost all cases.
Leakage per se does not shorten life. The secondary effects of leakage, like heating, may do that. Leakage which is notably worse than you expected in the device is an indication that the device is damaged in some way. Leaky capacitors may fail soon. Leaky diodes may fail and also waste the power you're using. Leaky transistors the same. But if you have expected the leakage and provided in the design for the expected amount, leakage does not necessarily shorten device life.
HEAT shortens device life. Thermal cycling (hot-cold-hot-cold....) shortens device life. Leakage only as it contributes to those two.