To answer your first question, the TV should be sufficient. At 1080P, you'll be hard-pressed to find a monitor with better resolution.
If you are not playing any games or doing any graphical development, don't spend anywhere near $240 on a GPU. In fact, if your motherboard has on-board video, that might be all you need (if not, or its insufficient or incompatible, you really don't need to spend more than $50).
What Video-inputs does your Toshiba have available? Component (that yellow-RCA plug), S-VIDEO, Composite (red, blue, green RCA-plugs), DVI, VGA/D-Sub (like old monitors), HDMI?
If your TV is close-enough to the one you linked, you have your option of VGA/D-Sub, HDMI (DVI-adapter), Composite, and Component. I would go the HDMI route, either with a GPU that has an HDMI port, or using a DVI-to-HDMI adapter on any GPU with a DVI port (these's days, thats 99% of them).
If you want to use that D-Sub port, again, you just need a DVI-I or VGA output (DVI-I has analog and digital, so you can use a DVI-to-VGA adapter, and the stereo audio cable).
If you want to use the Composite or Component video inputs, you need a video -card that supports those outputs (and you'll need a cable that splits your stereo-output to Left-Right [red and white] audio cables). Many main-stream and value-cards come with these outputs.
Like I said before, do not spend more than $50 if you are only going to surf the web, play Farmville, and watch DVDs/Blurays. You could get away with basically any discrete GPU card available.