Cooling trays are wonderful things...have one for my laptop, and have never regretted it!
(When you buy one, remember to keep the fans clear of obstructions, and the intake vents not be blocked by anything!)
But, from the sound of it, and the way you're using (the laptop) (now), you have all but hopelessly clogged the wee puir beastie, which means that you'll have to have it cleaned. You should NEVER have the laptop on your lap, a couch, bed, etc. The lap is not anywhere near dangerous to the laptop (but it is to your reproductive potential if you're a guy) as the others.
Whenever you place a laptop onto a soft surface, it sinks in, which means the vents are blocked, so your laptop can only dump heat via radiating it. That is bad for the computer. Given the fact that you say it has overheated on multiple occasions, you have probably caused some damage to the CPU and possibly other components.
As for leaving the laptop always plugged in, no that in and of itself will NOT harm the battery (assuming the battery is/was in good condition when purchased). When your laptop is turned on, the voltage that the laptop doesn't use is shunted into the battery to "top it off" if it needs it. If it doesn't, it's not used. (The power brick is supposed to prevent excess electricity from reaching the laptop.)
That having been said, to keep the battery in good health, you *do* have to allow it to discharge periodically. If you don't, the total amount of power the battery can deliver actually shrinks, even if the total amount storable doesn't.
Have a PROFESSIONAL tech who does this for a living look at your laptop and figure out what is wrong. If (s)he can fix it, great! If (s)he can't, buy a new one and treat it better! (Yes, that means using the compressed air on it regularly.)
Remember...dust is an insulator. The dustier the interior of your laptop (or desktop), the slower the heat leeches away, which means the components are working at higher than optimal temperatures, and heat is your computer's enemy!