I'd change 'go by car', which is perfectly fine, to 'drive':
A: you usually go by car/drive, don't you?
The verb 'is using' is fine here:
B: Yes, but the car belongs to my mother. She sometimes needs it. She is using it today to take Varun to the doctor.
There are no errors here:
A: I think I will take this bus. If I wait for any other, I may be late. If I am late today, my boss will surely be angry with me.