Is it permissible to take a mobile internet device (hotspot) that has been unclaimed for about 6 months? And is it permissible to take its battery, knowing that its owner has disconnected the internet service from it?

The aforementioned device falls under the ruling of a lost item (Luqatah). If it is something that people of average means would care to search for, then it must be announced for a year. If its owner comes forward, then [the finder] must return it to him; otherwise, the finder owns it, and must return it to him if he comes later. If it is something that people of average means would not care to search for, then the finder owns it from the moment he picks it up. Based on this, if your friend came to own the device, it is permissible for him to give it, or a portion of it, to you.