definitely i will shorten hard drive life span and also other components which is attached near to hard drive. My advice is to take it to the shop where you bought it. May be the hard drive is faulty or cpu case fan is not running well.
Short answer is YES , try a usb fan to help with cooling and consider a bigger tower, I used to have a mini tower pc and found a similar problem with mine so I bought a usb fan and helped alot for very little outlay
never , small case size of machines will not affect the hard drive life span and as well as getting hotter. it is the weather and continuous usage of system without interruption. try give some rest to our system then automatically the hard drive becomes cool
It will all depend on how big your tower is so if there isnt much space and all of the other parts of the tower will be hot and it will make it hotter and hotter, But if you bought it like that then you shouldn't have a problem really because its being made like that! I suggest you get a new hardrive or contact the manufacture