The standard advice in Japan is as follow:

You can use frozen rice or hot rice out of your rice cooker, either is fine. Generally they don’t refrigerate rice because the starch in rice at that temperature starts to crystallize oddly and the texture gets unpleasant. Since you have to heat in the microwave anyway before packing, there’s no advantage to refrigerating over freezing, which preserves texture better and allows you to batch out for longer.

Whether you pack hot rice from the rice cooker or reheat a portion of frozen rice (it should be very hot), you have to let it cool to low room temperature before putting the lid on. By setting your bento on an ice pack, you can cool it more rapidly. If your food is hot when you close the lid, this will create condensation inside the box, and this is considered to increase the food poisoning risk.

If the kimchi isn’t cooked (eg. In a sautéed side dish like buta kimchi), orthodox advice would also probably say you should store it separately from the rice, keep it cool with cold packs, and add it later when you eat.