what is the smallest number of calories we should NET?

The official answer is 1200. Realistically? I think it's ok to sometimes have a net of 0 IF AND ONLY IF you've had a HUGE workout (like over 2000 calories). So long as they pan out ok as an average don't worry. I personally am a bad example bc I workout a lot and don't eat enough but I net 300-500 on workout days. I would not recommend anyone should do this. It makes me grumpy and tired.

It really depends on the person and how accurate all of your calorie counts are. It is recommended to not go below 1200 calories net per day but fluctuating low and high day to day can be helpful to some people when loosing weight. If you body doesn't have enough food it stores it as fat as a protection that we have again starvation. Play with the counts and figure out what works.

If I net less than 1200 I get shaky and have zero energy (which in turn makes me crave something with lots of sugar)

I only very rarely net less than 1200 and have not done this for ages as it makes me feel so crappy

1200 is arbitrary. MFP takes into account your activity level, which is also subjective. You burn more calories than what you spend actually "working out," because it takes calories to make your body run. Your gross count is therefore inconsequential (except for of course the TYPE of calories you take in) because theoretically you are working it off - so your net is your calorie level excluding exercise and calories replacing exercise.

With this in mind - your net needs to be enough to cover the OTHER activities your body goes through every day - breathing, driving, walking, working, digesting, pumping blood, everything your body does to make you function. A net of zero would leave you with nothing going in to cover this, which is where serious damage can be done. While 1200 is a good guess, your lowest net depends on YOU, and only talking with a real nutritionist can help you find the answer to that! Don't put your body's safety solely in the hands of anything but common sense or actual experts, please, for your sake!

I think you are asking the wrong question.
Trying to get away with the smallest amount of calories possible sounds like a one way ticket to an eating disorder to me.
Better to ask "what is the RIGHT number of calories to net that allows me to lose weight slowly and safely?".

To answer your question, just look at what MFP recommends when you set it for a 1 pound weight loss a week - try that for a month and adjust it if you need to after that time.
Or see a professional to get good advice on being healthy, you state that as a goal, so don't try to be unhealthy and eat too little.

The official answer is 1200. Realistically? I think it's ok to sometimes have a net of 0 IF AND ONLY IF you've had a HUGE workout (like over 2000 calories). So long as they pan out ok as an average don't worry. I personally am a bad example bc I workout a lot and don't eat enough but I net 300-500 on workout days. I would not recommend anyone should do this. It makes me grumpy and tired.

Sure, anyone can have a low eating day very occasionally and get away with it, that's just life. But saying that 0 net calories is OK???
I don't think so.

Jesus. Read what I said. If it pans out to a healthy weekly average yes netting fine is 0 in exceptional circumstances ie very long hard workout day, like if you ran a marathon. You're just saying exactly what I said but trying to make me look bad. What's your problem?