You could draw the solder back next what remains of the pads. No you can't just connect the pins to each other as they are for different voltages. So all pins need soldered individually
It's sounds like there's an issue with the mainboard. Could be the hdmi chip or a processor issue. Try something other than the roku stick like and dvd player with hdmi output. I have 1 I'm working on same issue when chromecast is plugged in but fine with sky hd box
the grease is thermal paste its supposed to be there unless you want to add new paste, as for the fan you should be able to see it when the bottom is removed. use tweezers to pick out the fluff
depending on where your located theres companies around that can extract the data from the memory chip. other than that you would need to get it repaired for data recovery
when you replaced the battery did you connect the battery before the screen, if so you have probably caused a backlight issue. shine a light on the screen to see if you can see an image
