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Your car alternator can provide more than 1,000 watts. A 100-200 watt load won't hurt a thing.

I'd seem into either a yellow prime optima battery or kinetic battery. They're made for automobile audio and have can last longer with out the automobile walking and rfear of walking the battery useless.

Yeah, that may be too much for your alternator to handle. Unplug the cooler when using the inverter, and the car needs to be running while any of these are being used

The cigarette lighter is usually only fused with an 8 or 10 amp fuse. 100W continuous is max for the lighter. Your laptop alone will use that much.