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Sure you can use the laptop without the battery and plugged on A/C power! =D
But my friends always ask me to put back the battery before shutting down the laptop.
No idea why too, but there seems to be no difference shutting down without the battery. (But I play the safe way 
)
While you're running on A/C power try to put your battery somewhere cool and dry. (Heat is battery's biggest enemy!)
1 disadvantage of using A/C power is that you would lose your un-saved data in the case of power failure 
I'v done the same method as you've mentioned above.
But just remember to charge it at about 20% or so and you would be fine.
I admit sometimes I drained my battery (because there was no a/c power available), but it decreases battery lifespan faster than charging at 20%.
Hmmm.. Can you ask your friends about the battery issue when shutting down? Haven't heard of that before. 
Other than losing the unsaved data when power failure occurs, will there be any other negative effect on the laptop? I'm currently using the laptop w/o the battery because I use it on long hours (like the whole day or more, w/o shutting it down) and I'm afraid if I don't remove the battery and just let it plugged, it might decrease the battery lifespan.
And what do you think should be better? Just use it plugged w/o the battery or do the cycle (unplug when full then charge it again when battery comes low about 20%) ?
Haha sure I would like to know the reason behind it too lol!
There will definitely not be any negative effect for long usage of A/C power supply to the laptop. (Of course natural wear and tear still takes place)
Just plugged it in w/o battery. Do the cycle about once a month to ensure that your battery still functions properly
