Because there is so much competition, rates will tend to vary only slightly. So it pays to shop around. It is also a good idea to take points with your mortgage as this will help bring down your interest rate. Plus you can deduct the points from your taxes the year you buy your home, so you don't have to wait 15-30 years to deduct it from the interest you pay spread out. Keep in mind that if you do decide to take points on your mortgage, that money is due up front at closing.
Closing costs can become very expensive.
The more questions you answer the faster your paperwork can get processed, leaving less of a chance for your paperwork to be returned to your loan officer as incomplete. Even if you are not using a down payment, you might consider putting one down. The lender looks at this and sees that you are putting some monetary responsibility in right from the start. Know your upfront costs. Put aside the money so that you have it for later.