Combine oil, garlic and crushed red pepper in heavy small saucepan. Stir over med.-low heat until garlic is holde, about 5 min. Remove from heat and let stand at least 1 hour. (Can be prepared a day ahead. Cover and refrigerate. Bring to room temp. before using.)
Pour 2 cups warm water into large glass measuring cup. Sprinkle in yeast and let stand until yeast dissolves, about 10 minutes. Whisk in 3 Tbsp. garlic oil. Combine 2 cups of the flour and t 5sps. salt in bowl of heavy duty mixer. Add water and yeast mixture and beat until incorporated. Mix in enough remaining flour to form soft dough. Beat on low speed just until dough is smooth, about 3 minutes.
Brush large bowl with 1 Tbsp. garlic oil. Transfer dough to prepared bowl. Turn to coat. Cover with plastic, then damp kitchen towel. Let dough rise in warm, draft-free area until doubled in volume, about 1 hour.
Place chopped tomatoes in colander set over large bowl. Toss with 1 Tbsp. coarse salt. Let stand 35 minutes. Rinse under cold water. Transfer tomates to paper towell; drain well.
Prepare oven to 450 deg. Brush 15-by-10-by 1-inch baking sheet with 1 Tbsp. garlic oil. Punch down dough. Knead briefly in bowl. Transfer dough to prepared sheet. Using oiled hands, stretch dough to roughly fit pan. Press dough all over with fingertips to dimple. Sprinkle rosemary, then tomatoes, pressing some into dimples. Sprinkle with basil and remaining 1 Tbsp. coarse salt.
Bake focaccia until golden brown, about 30 minutes. Transfer to rack. Cool. Cut into squares. Servew with remaining garlic oil.