Well...first of all, there is no direct link between how many times a day you move your bowels and colon cancer. Since the bulk in your diet has decreased, it's only natural that the amount of fecal matter to be eliminated would also decrease. As you move through the different phases of low carb towards maintainance, you should be gradually increasing your daily carb allowance and using mostly veggies and some fruits as well as nuts and seeds to achieve that increase. By the time that you get to maintainance, you may be eating more veggies (and a greater variety of them) than you ever have in your life along with some fruits and whole grains as well.
Recent studies have shown that there is no direct link between the consumption of animal proteins and colon cancer and as for the diverticulitis, it's more likely to be aggravated by grains than beef. Same goes for IBS.

Edited to add: I moved this post to the General Health forum in hopes that more people would see it and be able to add their knowledge and experience to the topic.