No. If you are in the intersection box when the light turns red, you have just committed a moving violation and can (and should be) ticketed for it. Despite popular belief, you are not supposed to enter the intersection unless you can proceed through it before the light turns red. DMV driving testers will fail drivers that drive that way.
No, you are definitely supposed to enter the intersection when making a left then turn on yellow or red. You will get dinged on your driving test if you do not do so. At least in California, maybe it varies by state. But that is how you are supposed to do it here.
To quote a CHP officer
"If a person is driving straight through an intersection (NOT TURNING) law requires a motorist to enter the intersection, only when there is enough room to completely traverse the intersection. For drivers making a left turn, this rule does not apply! If a driver faces a green light, they are authorized to enter the intersection and wait for clearance to complete their turn. HOWEVER, the driver is not allowed to enter the intersection for a left turn if they are facing a yellow light."