Could I exercise with a hangover?

Yes. Most people with hangovers don't really feel like exercising, but there is no contraindication to it. Better, since you are concerned for your health to the extent that you don't want to miss working out, don't drink so much that you get a hangover next time. Safe drinking limits for women have been established as no more than 7 per week, and no more than 4 on one occasion.
Yes. Drink plenty of water to be sure you are hydrated before beginning. Everything may feel more strenuous than when you are not hungover; this is your body protesting what you have done to it!