Does a fully-charged battery supply power to a laptop/notebook if the mains is connected?
Does it behave like a mobile-phone (which needs a battery)?
Will the battery-life be improved if it is used lesser? (Li ion)
To answer your specific questions (as opposed to just chiming in and disputing other answers)...
Only if the power coming from the DC adapter is interrupted or reduced past a certain point. If the DC adapter supplies close to the requirements of the laptop, the battery will not discharge while the unit is running on current being supplied by the mains. You can easily test this yourself by replacing your adapter with another of a similar wattage. For a more concrete example, a Toshiba Satellite A75 requires a 120 watt adapter (19v 6.3a) but you can use the laptop with a 90 watt adapter. You must take certain steps to lower the power drain (reduce screen brightness, lower processor speed, etc) or when the laptop has to work hard, it will kick off the 90 watt adapter, and start using just the battery. Otherwise, if the proper DC adapter is being used for a laptop, and there is no break in the flow of electricity, and the DC adapter does not fail, then the laptop will not use the battery. You could remove the battery if you so decided, and still use the laptop. This is not disputable. This is a proven fact.
Again, providing that you are using the proper DC adapter, and there is no interruption of electricity to the DC adapter, and the DC adapter does not fail, a laptop does NOT behave like a mobile phone. It behaves like a desktop computer.
Think of it this way. Do you have a UPS for your desktop? You know.. an uninterruptible power source? The battery in a laptop functions pretty much exactly the same way. With a UPS, you plug it into the wall, and you plug your computer into it. As long as there is power from the wall, you never use the battery part. You only use the battery part if there is no power from the wall. But... if you remove the battery part, and you kill power from the wall, you don't use the computer.
Search the web and you will find experts of all calibers giving varied advice on this. NiCad batteries had a "memory" which worked in such a way that the battery had to be fully discharged before being recharged. LiOn batteries do not suffer from this. That said, you should expect about the same lifespan length as an average hard drive... 2 to 4 years. This is not to say you will get optimal use after 2 or 3 years. This is to say it will still hold a charge for some actual measurable amount of time (even if it is only 20% of original capacity). Will using the battery less extend it's lifespan? Of course. LiOn batteries have a limited number of available charge cycles. They will hold less charge after 250 cycles. They will retain less charge after another 250 cycles, etc. The fewer number of times that you charge/discharge the battery, the longer it will last. This article specifically tests smartphone batteries... but you get the gist. Read up. It's interesting.
How do I know what I know? I've taken apart and repaired more laptops than most people have ever seen... and that includes people who sell them at the big stores (seeing them, I mean). One common diagnostic step is to power the "laptop" without the battery. This also includes removing the motherboard and powering it. In some cases when monitoring heat bleed on older boards, this would mean powering a motherboard for hours or even days without the battery.
All HP laptops and IBM I have used work without the battery. And when connected to the mains the battery charges but is not used to power the laptop.
As for the battery-life, if a li-ion is unused for a long time it will lose capacity. Optimal storage charge is around 40%. more info here
God, will these twenty year old battery myths ever die?!
Of course not!
Look, with the battery in, the battery manager can do whatever it thinks is best for the battery. It can float the battery, it can charge the battery, it can discharge the battery.
With the battery out, the battery manager can't do a thing. The battery gets what it gets.
How could it possibly help to remove it?
The only thing I could possibly imagine is if the battery got hot in the laptop. But that would indicate a pretty crappy laptop design.