If the temperature in your car goes from blowing hot to cold, then I would lean towards the heater core (or some other part of the A/C).
A bad thermostat would cause overheating, as David suggested.
And an O2 sensor would affect your mpg, or nothing at all, depending which one. Like Snackmaster said, get any and all of the "error codes" read at a local auto store and post them up here. We can better diagnose your problem(s).
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