There isn't too much in the price really. I can get the 12v lighter socket chargers really cheap from China, but we all know what the quality is like!
I think you just answered your own question there.
If I use an inverter, will I need to keep the engine running?
You won't need to keep the engine running but, if the engine isn't running, then, of course, you will be running your battery down.
Depends on how your 12v socket is wired. Some newer cars kill the juice to the 12v sockets when they aren't on or aren't in the AUX key position. You'll need to check with your manufacturer/model of your car.
Make sure the inverter is large enough to power the laptop (power = volts x amps). If your laptop uses 2.5A @ 19.5V then you need at least a 50W inverter. A bigger inverter would be a waste and may use more power and run your battery down faster.
A 150-200W should do swell. A little headroom if you'd ever need it for another device, and not overkill. Also, OP, please do not go overkill here either and try to run something that will fry your 12v socket wiring. You don't want to be drawing more current than the vehicle's wiring can handle. I've seen wire damage occur when the fuse didn't properly blow on a 12v socket. It's not pretty.
If you need a higher wattage, you'll want an inverter that is wire direct to the battery with properly gauged wiring.
Less expensive inverters produce a MSW (Modified Sine Wave) while more expensive inverters produce a PSW (Pure Sine Wave). Some people report that their devices run hotter with an MSW. I've used MSW inverters in my RV for years and haven't had any problems.
Once again, you get what you pay for. I wouldn't be running a $500-$1000 laptop on a $5 inverter. Probably a little exaggerated, but you get the point.
On a related note to this whole ordeal, may I suggest a separate deep cycle marine battery to use with the inverter? If you're running the vehicle, great. You don't need to worry about battery drain. But if you want to be parked with the vehicle off for more than 30-60 minutes, I would go with an auxiliary battery. Back when I was doing car shows with my '84 Camaro, I had dual monitors setup, along with my PS2, and speakers/sub system and it ran for 3-4+ hours on a charge. I'm sure it would have kept going, but I didn't need to run it any longer. I'd imagine a laptop drawing that little bit of current would run for substantially longer.