Pretty much correct. The respiratory system doesn't blow off any glucose, however.
The issue with DKA is that the body can't use sugar for energy (due to the fact that insulin isn't there). The patient will continue to eat sugar, however. The body, in place of sugar, will break down fats for energy, which leads to the presence of ketones in the blood. These ketones lead to a metabolic acidosis (hence the name, "diabetic ketoacidosis"). The body, sensing this imbalance in pH, attempts to compensate by blowing off excess CO2 (here is where you get Kussmaul's respirations). That's why you should avoid RSIing a patient in profound DKA. If they can't hyperventilate, they'll become even more acidotic and you'll be working an acidotic arrest. Bad news bears.
When it comes down to the renal side, you're correct that renal failure is an option. My understanding is that the mechanism behind the excessive urination is that in the renal system, the body will try to eliminate ketones and glucose through urinary excretion. Glucose and ketones pull fluid with them when they are eliminated by the kidneys (oncotic pressure if I remember correctly?). This is where you get the polyuria and, subsequently, polydipsia. Electrolyte imbalances can then follow, which can be compounded by nausea and vomiting. DKA patients are sick and critically ill.
Treatment of these patients involves fluid resuscitation/rehydration, and correction of electrolyte imbalances. Insulin drips are started as well, sometimes only after fluid resuscitation is done. Insulin drips and fluids are continued until the BGL reaches a certain level, then sugar (for example, D5W) is added to the fluids so the insulin drip can continue so that the anion gap can close. Once the anion gap has closed and sugars are under control, the patient should be switched to sliding scale insulin.
Management of these patients isn't necessarily easy, and they need to be monitored very closely. Electrolytes (especially K+) need to be monitored throughout treatment (remember, insulin drips lower potassium levels, and may lead to hypokalemia). The whole process is actually fascinating.