Unfinished story carry forward to next sprint

What have you committed to? Are you able to meet your agreed Sprint Goal, and to deliver an increment of release quality work?

By the end of sprint, we always have story half way in development or SIT, and have to be carry forward to next sprint.

A few observations:

1. Many of our story need more than 1 sprint (2 week sprint) to complete. 

Is that a must to size the story so it can be complete within 1 sprint? 

@Adriana Tung, The purpose of a Sprint is to create a Done Increment that is potentially releasable. Having said that have you along with your team been able to identify why you are not able to complete work within a Sprint? Are there any dependencies? Assume you have the ability to make changes, what changes do you think could help you complete them in within your Sprint timebox?

2. We often start our story in mid / end of the sprint, knowing that will not be able to finish within the sprint.

How can we complete all stories just right within a sprint? There're always a few days capacity left by end of the sprint to pick up a few new stories.

I hope my previous response helps answer the first half of the question. For the second half, if you have capacity to take on a few more stories, nothing is stopping you from doing so. As long as the Sprint Goal is not endangered (i.e. you must be able to meet your Sprint Goal) you can take on additional work if you have the capacity.