This is really your decision. Depending on what places you would like to visit, downtime, I would recommend 3 days at a minimum - 2 weeks. Did I tell you that there are amazing surgeons here too! Hello, this is Colombia! You will see very good looking people.
There are countless options depending on what your desires are. If you want to be out in the open and enjoy the beautiful scenery Colombia has to offer I would stay at a "finca hotel". Keep in mind you will be a bit farther away (15-30 min) from the city which is not bad at all. In that case, I'd recommend renting a car. If you want to stay in the city there are an excellent variety of hotel choices. I think Mocawa is one of the better places to stay in that case. You have taxis at all times available.
For every dollar, it is approximately 3,000-3,400 pesos depending on the exchange rate. Your dollar goes a long way. Exchange at the airport in the baggage claim section in Bogota, they do not charge commission fees there.
No, you do not. You do need to provide where you will be staying and answer why you are coming and for how long. You can stay for up to 90 days.