Does the length of time it takes to finish my medical treatment affect when I can settle my claim?

Typically, you should not try to settle your case until all medical care relating to the accident has been completed or at least ascertained.  If the medical care is going to go on for an extended time, then you will need to treat long enough for your doctor to be able to explain with reasonable certainty what future treatments will be necessary and what the treatments will cost.  (It would be very unfortunate to settle your case quickly only to find out later that you have to pay for a surgery down the road.)