Strange. My late 2008 MacBook can do it all without any issues or overheating.
Keep in mind though that buying a second/hand electronics is always a gamble especially if you don't know what you are doing. You should replace a battery as well as the thermal paste/clean the dust inside the machine etc.
Yes definitely, this machine needs a new battery and get clean because the last owner didn't do it since 2011. I think the real problem here is the poor mac optimisation and the "service battery" thing, It drains the battery really fast and I believe it overheats the machine faster.