1. It is very unlikely that the sink will fall into the cabinet unless they removed or damaged the support devices built into the rim. Which should never have happened.
2. The caulking may have gotten into the space between the side of the rim and the hole in the countertop where your putty knife can't reach it, making force the only way to separate it.
3. You may have to pry the rim up, but the problem with that is if the caulk is in the groove you may chip the countertop doing so.
4. You are trying to separate between the rim and the countertop, aren't you? You do not want to separate the sink from the rim.