Is running laptop in AC supply good?

I use my laptop mainly for playing games.I play games like crysis2,BF3,assassin's creed.I play for 4-5 hours daily.While playing I plug my lap to AC.The battery gets fully charged within 2 hours & for rest of time it's been plugged to AC.Also it gets heated up while playing games.So I wanted to know that:
1.Should I remove battery & plug my laptop to AC while playing games?
2.Will everyday removing & inserting of battery affect my laptop or battery?
3.If suddenly power fails then it's sure that laptop will shutdown.Will it affect laptop?
4.Is there any suggestions for a low budget cooling pad for laptop(HP Pavilion dv6-7012tx)?

How i understand it. (i could be wrong) The types of battery's that are used now it doesn't effect the the battery to leave it plugged in all the time.

If you remove the battery it wouldn't have any effect either just the wear from taking it in and out.

If the the power fails and the Laptop is plugged into A/C power without the battery the laptop will shutdown. This can cause data corruption.

My suggestion is just to leave the battery in while leaving it plugged in.

You should WORK the battery every now and then. Let it drain to 50% or a little lower then recharge to 100% every week or 2.

Never drain the battery all the way and try not to drain it below 20% on a regular basis.

Other then that leaving it connected to the AC Most of the time is perfectly OK.

I think u guys didn't get my question.
When I googled for my doubt a lots of sites said that "while playing games laptop gets heated up to a high temperature & this continuous heat can affect battery's life.Also prolonged charging also affects battery's life.So remove battery & use laptop in AC while playing games."

Is this true?

1) No you should not remove the battery. No reason to. If the system get so hot that it can effect the battery then the rest of the system will die before the battery. Heat is the killer of all things Electronic.

2) Yes. Read # 3

3) Yes by running the system on AC only without a battery inserted and if the power goes out the whole system could become corrupted. So don't remove the battery. No good reason to.

4) I have no suggestion.

Thank you guys.