A car is your best option, driving the coastal area is not difficult at all, I have no problems driving it despite being used to driving on the left.
'The White Hill Towns' is not a specific place or area. Pretty much all houses and buildings in the Costa Del Sol are white and any town or village will consist of white houses.
Your trips are long and far too many to cram into a week. You also don't say when you're going, if the family includes children (and their ages) and how many days you actually have there (does the week include arrival and departure days?).
I would leave out Morocco. Tangiers is not an attractive or interesting place and the ferry journey consumes a lot of time (plus getting to and from Tarifa). I think most people visit just to be able to say that they popped over to Africa whilst in Spain. Gibraltar is not too long a journey from Mijas and it's reasonably interesting. I found it to be a bit like Britain in the 80's but the rock and the tunnels at the top were interesting enough. I can't get excited about being surrounded by the familiarity of M&S, Next or Tesco however.
Unfortunately Mijas Costa is not ideally situated for exploring the best of Andalucia. The Costa Del Sol is a beach resort area, frequented by predominantly Northern Europeans seeking summer sun and beaches, most of whom never venture further than their hotel or the strips of bars and restaurants that surround them. Granada would be the maximum distance I would want to drive, mainly because my kids get incredibly cranky after 1 1/2 in a car but also after a long day of touring tiredness would be an issue for me on the return drive particularly with somewhere like Ronda where the drive through the mountains requires absolute attentiveness.
I would look at trips to Nerja, Malaga, Granada and perhaps a quick trip to Marbella Old Town. For food I recommend visiting the hill town of Benahavis, full of fantastic restaurants.