Overheating and freezing Macbook

Hello, I'm owner of macbook pro mid 2009. Until few days ago I almost never had any problems with it. About two weeks ago it began to overheat (about 80 C while idle, usually it was about 50-60 and it gets over 70 right after I log in, without any programs open and goes higher) and sometimes it does not wake up after opening lid. Fans begin to spin but beside that nothing else happens, the screen stays black. Restarting helps but only for a moment. Also, while power adapter is connected, it sometimes doesn't charge without adjusting the plug (recently I noticed that my adapter was damaged, but it still happens after replacing) or computer shows charging but after few minutes suddenly crashes and battery shows 0%. I've tried resetting pram and smc a couple of times already but I did not notice any changes. Apple hardware test doesn't work for me and I don't have original disk. Maybe it is battery's fault, it's seven years old and I'm going to replace it in a days, but I don't think that old battery would cause all of this. Do you have any ideas or suggestions on what to do? Thank you in advance for any help.