Which is the best mortgage to get if I don't intend to stay longer than 5 years? 15, 20, 25, or 30. Goal is to minimize interest and pay down principal as fast as possible while being able to pay other bills? What is the most popular mortgage and why? Thank you.

The best for you is to have a 15 year mortgage, you build equity faster and if you sell in five years is is a selling point to have a mortgage with only ten years left on it, if they assume your mortgage. Plus it is nice to paay off the house before you get to be sixty years of age. I have done two this way and still young enough to enjoy life.

it depends on what your long term goal is. Are you looking to sell the place after 5 years or are you going to refinance again?
If you are looking to sell the place, you shouldn't worry about paying down your loan at all. Your home will gain tens of thousands of dollars in equity it these next five years, but you will only pay it down a few thousand dollars if you get a good interest rate and make extra payments.
If you are looking to refinance, then either take out an interest only option or a deferred interset option and invest the rest of the money for these 5 years in an account that can earn you 10-12% interest. At the end of the 5 years, you can take that money out and pay down your loan if you want, or u can put it to better use like re-invest it.
you also mentioned that you want a lower interest rate. Banks will give you a lower rate based on the type of loan and other qualifying factors. First of all, a 30 year fixed will have a higher rate than a 15 year fixed. An interest only program will have a lower interest rate than either of those, and even with this option, the rate fluctuates depending on the number of years the rate is fixed. You can get a 5 year fixed option, or you can get a 1, 2, 3, 7, etc. it's up to you. Keep in mind that just because you're not paying down your loan every month, doesn't mean that you can't pay one big lump sum at any time you choose.
speak with a professional that can explain it to you in person or over the phone, or email me if you have any further questions.