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Will a screensaver run my laptop's battery down?

They will use your battery power. Keep in mind your monitor uses up quite a bit of power, so having it not just on but also displaying moving images will continue to use power.

If you're concerned with battery life, you should do one of two things to maximize it, 1) Have your monitor go to an all black screen until you're ready to use it again, or even better 2) set your settings so the screen turns off instead of displaying a screen saver.

Truth be told, screensavers were something of importance years ago as an attempt to minimize potential image burns into screens and monitors. Today, that threat is pretty much entirely gone, but because so many people liked the moving images, it was simply kept, as opposed to an all black screen or having the monitor disabled until activity began again.

But generally speaking, yes having your screen on will continue to use power, having it display (moving images) will use even more power. Set it to go black or disable the screen for short durations where it's not being used, and for longer periods, have it go to sleep or hibernate.

How do you charge a lithium ion laptop battery?

Use it mostly on charger. Try to not go too often deep discharged, if possible. I used this approach and my laptops (I have three now) goes first at tenth year, second beyond sixth year, they are obsolete now and still have 60% respective 65% of battery life.

I had used the last netbook on charger day-by-day, around 15–16 hours a day ON, part of this time was of course in standby but plugged in, and I stopped (unplugged) it when going to bed. It does not matter too much if you forgot it on charger, but yes, there is a (very) slightly damaging effect of living it on charge all the time. But it is FAR less than cycling it.

I did less than twice a year full deep discharge. I avoided it, because deep discharge cycles counts more to the lifespan. These are only necessary sometimes to reset the battery electronics which tells you accurately the capacity. But is not really need to be accurate - at least not always. The error of display percentage grows to maybe 10% in a year.

As a last advice, if you travel a lot and do not have where to plug often and also don’t have time to leave it on charger and work with it on charger, the least damage when cycling it is in the middle of the state-of-charge. That means if you charge it as soon as possible (even if its only discharged to 45–55%), to avoid deep discharges, and unplug it when goes up beyond 70–75%. As much time is in the middle, will have least damage from cycling. A 10–20% charge or discharge in the middle of the charge is not sensed by the battery life as a 10–20% of a cycle, but far less (maybe 2–3%). Yes, the electronics will count it as 10 or 20% from a cycle, but that does not matter in reality for the battery. But when you know you’ll have power for hours (when at home), you can keep it on charger to go to 100% and continue to keep it at 100%, because is less damaging to stay at 100% than to do microcycles.