Assume one of those online calculator determines I need a calorie intake of 1,600 per day to lose 2 lbs per week. Could I consume more than 1,600 assuming I'm burning that off in exercise (say I consume 1,900 calories -- but I run until I burn off 300 calories). Would I still lose 2 lbs per week?
It depends on what the specific site gives you: BMR, RMR, or RMR + estimated calories that you burn according to your lifestyle. Most websites offer the later two. If you're not working in a physical job, the best approach will be sticking to the RMR calculator and add the calories that you burn in your workout (100-150 for a strength session and whatever the machine says in a cardio session).
One thing to remember is that these calculators never reflect precisely the real needs of your body. Track your progress and change your total intake if things are going too slow/fast.
Obviously I don't know your bodytype, so I don't have exact numbers for you personally.
It all depends on how much calories you burn during the day. If you consume 1900 calories but you burn 2000 for example, you'll lose weight. You will have to find how much exactly you have to burn in order to lose the 2lbs a week if that is your goal.