If you can afford it, get the FE21B. It's a stonker of a laptop and I would give my eye teeth for it.:-) click here This has WiFi built in as well as a dial up modem. It would do all your jobs without blinking.
The FS515E is a massive leap over your dead computer and is generous of your insurance company. It is a good laptop but no where near the other one. This also has WiFi. This computer would also do what you require.
WiFi is a network that works by radio waves. Basically if you in a hotel, restaurant, train station, airport, office, etc and it has a WiFi network. You can, if you ask permission, some times have to pay, connect your laptop on to the internet. This would be fast broadband. You would not have to plug any wires into the laptop and the laptop could be moved around, within reason and you would still have a signal.
I am typing this with a WiFi laptop in my garden. No wires. The laptop is remotely connected to my broadband router. The system is secure and no one can enter it.
My choice would be the Sony FE21B.