Since you replaced the battery make sure it’s plugged in securely and the black connector is pushed all the way into the port on the mainboard. It may be the ac adapter/charger you are using, make sure the charger is fine as well.
Yes, a dead battery can cause power issues even if you are using a charger as power. I would open the unit and unplug the affected battery. You should be able to use the laptop with the internal battery unplugged and the ac adapter/charger to power it, but as with a desktop when you unplug the ac adapter/charger from the unit you will lose power. However, look into getting a battery replacement.
1. Have you checked the charger? Make sure the contacts on the side that slides into the DS aren't bent or broken. If this is an issue just replace the charger. 2. You may also need to replace the battery if the battery is not holding a charge. And if those two things do not solve the issue: 3. It's fairly common over a period of time for the charging port inside the DS to become worn out. Get a flashlight and take a look at the charging port to see if anything looks bent or broken. If this is the issue you would need to bring it to one of those cell phone repair shops (I would call beforehand to make sure they work on handheld gaming systems) otherwise you might have to buy all the soldering equipment and parts and do the soldering job yourself. Good Luck!
If you are using one of those ISP hotspots that require prepaid access or an account you would log in with your username and password for your ISP account after adding the wifi connection in your settings. A browser should popup asking your login credentials. Otherwise, if you are trying to connect to an open network just choose the network in your wifi settings.
1. When you go to settings and try to pull up wifi list does it show any connections at all? (yours, neighbors, etc.) - If no connections are shown make sure airplane mode is off and your wifi setting is turned on, otherwise you have a wireless card failure. -If you do see connections and attempt to connect to your network is it giving you an error? If its a password error and you know the password is right, forget the network (normally by holding down on the highlighted network and select forget) and add it again and try. If it is another error please post. -If you do see other's connections and not your own then make sure your 2.4 ghz bandwidth frequency is turned on, some devices do not have the hardware to see the 5ghz frequency.