I'm far from an expert but I'll answer from experience. First of all, i have seen people lose weight fast on atkins but unless you stick to the plan when the weight is off it just goes straight back on. This is the same for any diet. The benefit of atkins in my opinion is that you don't feel like you're starving yourself, i have lots of energy and eat when i want!
I've been doing atkins for a few weeks now. The first week I felt awful.. I think they refer to this as atkins flu. I was sick in bed for 4 days - I reckon though I had a cold and the change in my diet just made it worse. It'd didn't last though and I was good as new the next week.
As far as cravings are concerned I am the worst person i know. I have tried atkins and many other diets in the past and whatever it is I wasn't allowed (usually crips, cake and buttery toast) is what I would really crave - in the end I always caved to my urges and ended up binging and putting back on more weight than I'd lost. Big mistake! I think to overcome your cravings you need to have a big dose of will power but you also need to be in the right frame of mind. This time starting atkins I was very determined and I just stuck it out.. through feeling ill the first week and all. The second week was great, felt better, cravings were gone, and i actually had more energy than in a long time!! I will admit however that i have slipped up .. I ate a small bit of cake one day and then all my cravings came back and i spent the night binge eating. I felt so sick afterwards and the next day - physically and emotionally! It really did prove to me the importance of not slipping up ... it's just not worth it!
Exercise on any diet is very helpful. The more calories you burn the more weight you lose. If a diet requires no exercise (like some of these 18lbs in 4 days detox plans) it is likely because it is so low in calories that it would be dangerous to exert yourself - that's just my opinion but i can't see anyone saying im wrong ..
In terms of what exercise to do, if you want to lose weight I believe any sort of cardio is best ... walking/running/cycling etc. Sadly this is an area im lacking in though.
Well I hope I have helped and hopefully someone more experienced with atkins and exercise in general will chip in with their thoughts and fill in the gaps that i have left. Best of luck to you