Will a crockpot brown ground meat

Stirring and scraping would prevent burned bits. The onions and peppers release water as they cook the mechanical action of stirring and scraping pulls up the brown bits before they burn and incorporates them into the peppers and onions.

You don't need to add any water to the slow cooker whatsoever. Your recipe is somewhat similar to the one that I use to cook my lunches but the quantities seem larger than mine, much larger. I assume that you chop everything up into small piece, including the chicken. The slow cooker should not be more than 3/4 full, so reduce your amounts if the quantities are too big.

Chicken breast doesn't need more than three hours on the 'high' setting in order to cook (six hours on low); this cooking time leaves most of the vegetables crisp (courgettes and pumpkin soften, but bell peppers, carrots and onions retain their texture).

Running a kosher kitchen, I would not know about adding cheese to the food, but anyway I would recommend adding the cheese to the food prior to serving, not while it is being cooked.

If you wish to separate some of the vegetables, one technique which I have used successfully is to put these vegetables in a wire tray (I use the plastic punnets in which we buy fresh mushrooms): these vegetable are cooked like the rest but are kept separate.

If you do wish to add certain vegetables after you have already started cooking, then you are supposed to add 20 minutes to the cooking time (every time one opens the lid of the cooker, steam escapes and it apparently takes 20 minutes to obtain the same amount of steam).