Most of this generation laptops come with Lithium ion batteries. Deep cycle discharge - discharging the battery fully to avoid memory effect, before another recharge - is not required for laptops which come with Lithium ion batteries. Li ion batteries do not suffer from memory effect - a phenomenon which causes deterioration the chemicals inside the battery causing them to reduce its full potential. The only issue with Lithium ion batteries' is -its capacity gradually starts decreasing from the time of its manufacture due to high internal resistance - which means it has low shelf life.
Will it cause to overheat your laptop if its connected to power source continuously?
No. The laptops batteries come along with a shut down separator which prevents the batteries from overheating.
Which one is better for the health of the laptop when its not in use? Sleep, Shutdown or Hibernate?
The approach for handling laptop can be decided on
1) Duration not in use
2) Effects they cause to internal resources like memory, disk space etc.
3) Whether the laptop is fixed or mobile
If the laptop is going to be idle and will be used in say 30 minutes, put it in sleep mode. It draws little power and generates little heat so it wont affect any components adversely.Mac users can put the laptops in sleep mode for longer duration too.
If you work on 20 applications and want to see them as it is next time when you start the computer, hibernate is the only best option. It automatically saves the content in the memory to the hard disk. The laptop uses no power in hibernate mode similar to shutdown mode.
If you are away from the computer for a couple of days, you have to obviously shut it down. Shut down clears the memory and gives opportunity to install patches. The con is while shutting down, the hard drive and other components get more wear and tear and use more energy.
When you are moving with your laptop, its better to put it in hibernate or shut down mode but never in sleep mode to prevent hard disk crash.