The sensors are what allow the temperature to be read. Without those the machine will most defiantly do as you have described. That is why you hear the noise from the fan but cant get a appropriate reading because the sensor is missing. Do you know if the hardware is all original?
You've got a iMac that is very upgradeable. I had one of these and looking for speed and reliability i wanted a SSD AND a traditional Hard drive. With these iMacs you can open it up and install BOTH the SSD and your 3.5in HDD with replacing the CD/DVD drive in your imac and installing a hard drive caddy. They allow you to keep a big storage type device like your Seagate drive and use a SSD at the same time. Im sure there is a guide here on iFixit but you can get the jist of it looking on youtube. Just something ive found to be very cool considering Apple products. Look into it. Dedicated SSD's are far better than the SSHD's. This is the caddy: http://www.amazon.com/SATA-Drive-Caddy-Apple-SuperDrive/dp/B00M03RIN2
You can (if your careful) flip the screen all the way down and let it rest. Best way would be to have a weight to hold the partially flipped screen in the air while you un-screw the LCD's flex connectors metal cover.
First thing would be to give it a full charge with the supplied cables. If it charges and turns on your ok. If not, then you may need a new battery or try using a different power cable. What phone is it?
Try to power on the device when its plugged in. If the charging bolts shows up it should be able to turn on. If it does not, then the power button could be the issue. If power button works then id say a new battery should be installed.
Best place for replacement logic boards (motherboards) for iPhones would be eBay.com Just make sure you know what board you'll need. Carrier, gigs and model of the iPhone are all needed to get the correct board. Also make sure the seller is trustworthy and dont buy anything with iCoud on it.