[villagetinker]

You can rent concrete grinders at Lowe's and Home Depot, I believe they have smaller units that will be suitable for your project. I would do the entire concrete surface as the grinding will leave a surface that will be much different in appearance then the rest of the surface. Also, if you decide to paint or otherwise coat the ground concrete be sure to use TWICE the recommended amount of shark grip or equal so you do not end up with a slippery surface. I just had the lanai completely ground and repainted, so I am familiar with what was done.
The next thing you need to do is figure out why the concrete buckled, I would look for tree roots and if you find these cut them or you may have a continuing problem. A neighbor had this happen to their driveway.
If you decide to work on the door, check on the bottom, in many cases there is an adjustable piece of trim with several screws, that you may be able to loosen and adjust for additional clearence.
Hope this helps.