Hopefully, your only problem is a dead battery. Charging/Replacing it should fix this. If that is not the case, it could either be a hardware failure (motherboard, etc.), or a software failure (virus, operating system failure, etc.). Head to the troubleshooting page for more advice.
Your laptop settings might be running intensive programs or with a bright screen causing the battery to drain quickly, or there might be a hardware issue with the battery itself. Look to the troubleshooting guide for more information.
There could be many causes for this. The most common is that your internet connection itself is down. You may try resetting your router or calling your internet service provider to find out. If this fails and your internet connection is in fact working properly, it may be a result of your ethernet cord. Try a different cord or plug it in to a different port (in your router or modem, etc.) and see if it works then. If all this fails, the ethernet port on your laptop is likely broken. Look to the troubleshooting guide for help.