Without more info about the blue screen that you got in Vista before installing Win 7, here's about all I can contribute.The initial problem
If the screen goes out again, check the inverter. You can get a Gateway NV52 inverter for $20. The light for the display needs high voltage AC and the laptop gets DC from the battery or power brick. The inverter converts some of that to AC, steps up the voltage then sends it to the backlight to light up the LCD screen so you can see what is displayed. If you do end up replacing one beware. It can bite you, even when the battery is out and unplugged because of the manner in which it steps up the voltage.
The test for a backlight problem (inverters being the most common culprit) is to boot up the computer and shine a flashlight on the screen and see if it is on but without a backlight. This can be hard to see with laptops especially if you are looking at a boot screen that is predominately black. Ironically, a blue screen is easier to see with a flashlight but the last thing you want to see is a blue screen
You inadvertently did a lot of the things that you need to do to get a bad inverter going again. You opened the case, probably wiggled it some, maybe reseated it. You also removed all power by unplugging it and removing the battery for long enough that any power in it holding it in a failed state seeped away.The thermal compound conundrum
I don't think that you should have put thermal compound on the chips but what's done is done. Laptops are very specifically designed for cooling. I don't think you hurt it and probably don't need to open it back up to remove it unless...
1. You gooped it on. If it's a very thin layer it is probably okay.
2. You monitor the temperatures with something like this http://openhardwaremonitor.org/
and see that the chip temperatures are too high as defined by the chip manufacturers specs (see their individual sites)
3. You replace the fan or inverter. Your in there anyway so you should clean it up.
To clean it use a Q-tip with a small amount of isopropyl alcohol and be gentle and careful because it's easy to smear it all over the place
Let us know if you have any further issues. Disassembling and reassembling a laptop and having it still work is no small feat. Congrats!