I can see how this would drive somebody crazy!!
Here is your order with my comments:
We lived in Pretoria for a while before we moved to Durban - I agree
We lived in Durban for 5 years - I agree
We were going to move to Port Elizabeth in 1983 but my job there fell through - depends on when the speaker was in Johannesberg, Pretoria and Durban i.e. before, after or around 1983. Are we supposing that they are speaking from a point in 2008?
We live in Johannesburg - I agree
We're going to live in Capetown - this might be interchangeable with the sentence below
By the time we move to Cape Town, we'll have spent 3 years in Johannesburg - this might be interchangeable with the sentence above
After we've lived in Capetown for a few years, we''ll move back to Johannesburg - I agree.
I'd love to hear whether your tutors agree with my interpretation!!