Can there be problems if I leave my old android phone alwys plugged to electricity?
Almost all concerns with phones remaining plugged in indefinitely stem from the health of the battery. Mentioning that the device is old and has a worn battery (and odn't want to deal with charging it constantly) means we can ignore those potential challenges. Refer to the question that user "dr01" linked if you want more info on that.

As for damaging/overheating your phone - that depends on your device and what your usecase for it is. Using it as a mini media centre and router will make sure it's at least warm all the time - but shouldn't cause any damage to it if the device doesn't have a history of being hot under normal use.