Diverticulitis....no. Diverticulitis occurs when a small pocket on your colon ruptures. I would advise against taking anything that might overstimulate the colon during an acute episode of diverticulitis, either from above or below.
No. I wouldn't advise internal clean songs under any circumstances. They can be risky, in fact.
Generally. If you are drinking adequate fluids and eating a high fiber diet I do not recommend colon cleansers as a routine except when preparing for an xray or colonoscopy or surgery. See my tips for proper diet and fluids. Do not waste your money on colon cleansers, they only make those selling them rich.
Generally - yes. Depends what the actual solution is you are considering. As a broad category - yes. We do this routinely to prepare patients for colonoscopy. Fleets phosphosoda, for isntance, is generally avoided these days due to concerns with electrolyte imbalances. You should speak with your physician about the solution you are considering using.