In a heavy large saucepan, bring all ingredients to a boil. Simmer to a thick syrup consistency, about 30 min. Cool.
Boil apricots and water. Remove from heat. Let stand until fruit is soft. Drain apricots and pat dry. Place in food processor. Add sugar and egg whites and process apricots are minced. Add 1/2 cup coconut and process until minced.
Transfer mixture to a medium bowl. Stir in remaining coconut. Freeze until firm.
Wrap heavy-duty foil around outside of 9-inch springform pan with 2 3/4-inch high sides. Spoon coconut mixture into pan. Spread over bottom and 2 inches up sides of pan to form crust. Bake crust until golden brown, about 15 minutes.
Using electric mixer, beat cream cheese, sugar and salt in large bowl until smooth. Add sour cream, rum, lemon juice, and vanilla and beat until well blended.
Add eggs and beat until blended. Transfer to crust.
Bake cheese cake until brown, and firm in center, about 1 hour. Cool cake in pan on rack 10 minutes
Cut around pan sides to loosen cake. Release pan sides. Arrange berry halves over top of cake. Garnish with mint, if desired.
Serve with strawberry sauce.