This question applies to laptops, but also to smartphones or anything that's battery based and rechargeable.
Does depleting it completely (Computer won't even turn on, or turn on but shuts down immediately) do damage to it? (Both the device and the battery)
Does it shorten the battery's lifespan?
Discharging a battery to a very low level can reduce it's lifespan. This not that big of a problem with modern batteries but still should be avoided.
Anyway this won't happen in your examples. A laptop or smartphone will stop working long before the battery is on such a low level. You even should do this from time to time as the measurement of your battery capacity gets less accurate over time. If you ignore the warnings of low power you will notice that it keeps running some time even after 0% (especially older batteries). If you keep it running until it shuts down itself that low point of the battery power is recognized and will be used from now on.
Li-ion batteries are based on chemical compounds that will work well for about 3 years, after this time the battery will start to degrade quickly. So you can try to handle your battery gently, but probably it won't help much.
Also, batteries are estimated to work for 500 charge-discharge cycles. It's sometimes said that to minimize battery's degradation it should be discharged to only about 20% and charged to 80%, but in that case you're only using 60% of battery's capacity, so you have to charge it almost twice more often, thus exceeding the 500 charges limit sooner.
