ok, good
Ok, so its flat on the table. I, would consider that bad.
I used to keep my laptop flat on my computer table, until I noticed that it was getting so hot. Solution, I bought a cooling rack for it to sit on, mine cost about $15. It definitely runs cooler now. My cooling rack doesn't even have a fan built in, though some of the more expensive ones do. And yes, the battery can put out a lot of heat, even when its not charging. But especially when its charging.
Did you know that heat is one of the TOP killers of laptops; along with drops, spills, and dirt.
And to answer your other question; yes you could run a laptop without its battery. BUT, its really better if you don't. On some models (like my Dell) the manufacturer highly recommends that I run operate with it installed. So you need to check with the manufacturer before you run without the battery installed. Some may even void your warranty if you run without a battery installed.
However, consider this: your working on the big project, or a crucial part of a game, or a etc...
And the power cord pops out, NOOOO! Your laptop is now completely OFF! If you have the battery in, the computer will stay running. And thus, no lost "whatever" you were doing.