Hello lilou.crapounette!
It sounds like you may be having some battery issues. How old of a device is it? If the device is over a year or two old, it might be time to replace your battery! The lithium batteries start to wear out around that year or two mark. Early signs of battery issues are shorter battery life, overheating, and even expansion of the battery, causing the screen to pop out of the frame. The shop you took your device too, do you recall if they were IFIXIT Mastertech Certified?
This would cause your heat issues. Now lets talk about your home button. I have repaired dozens of iPhones, and have noticed that some have wiggle with the home button. The easiest way that I have found to stop the wiggling home button issue is to first power down the device, then take the screen off with your anti-static mat or wrist band. And remove the home button retention plate. From here, you need to examine the plate and see if it is bent. If so, bend it slightly so there will be more pressure on the home button to prevent any home button wiggle.
I hope this information has helped you with your issues. Thank you for posting to the IFIXIT Forums!