Answer to: Why does my connect keep switching on and offCould be a power related failure by the sounds of it either the cord or the supply itself. Had an issue similar to this on a pc. Had to replace the power supply to fix it. If you know someone with a connect like yours, you can try to swap the power supply and see if that gets it working correctly. Could even go to a local gaming store that sells used parts and pick one up on the cheap just to test it.
Answer to: Battery overheating after little useBeing that that is a smartphone, it's possible there are too many programs running. In cases like that, even Galaxy phones will start heating up. Do you have wifi or bluetooth enabled and does it do this if you restart it then talk on it? Try opening a task manager and ending some programs, then talking on it to see if that helps. As for the charging, it is normal for it to get warm when plugged in. It is NOT normal for it to get scalding hot to the point you can't touch it. There could be a possibility the charger is faulty too and sending too much current. Try using another from a reputable brand, or borrow one, and see if you still have heating issues. Is the charger the original, or is it from something else?