Doc said if there's ligament/ tendon tear or rupture at ankle after sprain, I can't walk at all. Since I can walk, it means strain. No need mri. True?

Depends. If you are improving the tendon injury was probably mild and is healing. If you are not getting better after 3 months the tendon injury may be more severe. This still doesn't necessarily mean you need an MRI but may indicate more specific care of the injured area is required.
Not true. The ability to walk has no relevance on the severity of your injury. An MRI would help your doc make a more definitive diagnosis.