The urbanizations make good bases for a holiday, but if you want to see the 'real Spain' then you would do well to use your rental apartment only for sleeping. There is not much very Spanish about the area immediately south of Torrevieja, which incidentally is known as the Orihuela Costa.
Europa is a good car hire company. They are reasonably priced and fair with the fuel, giving you a full tank when you pick it up and expecting one when you return it. As you are coming in August it would be sensible to pre-book a car online.
The Orihuela Costa lays about midway between Alicante and Murcia (San Javier) airports. Driving time from either is about 45 minutes. Coming from Alicante and driving south, the first large town that you will pass will be Santa Pola which is only 10 minutes or so from the airport.
Santa Pola is worth a visit, as is Elche which is about 10 Km inland. You will also drive past Guardamar del Segura and La Mata, both towns are predominately 'Spanish'.
Within an hours drive of your base you have Lorca, Cartegena, Mazarron, Murcia and Orihuela itself.
There are obviously going to be days when you don't fancy a long drive, and nights where you prefer to eat more locally. Torrevieja town center has a pleasant square and marina, and plenty of good restaurants (a fair share of bad ones too!!). The beaches north of Guardamar are far less crowded with much less chance of being hassled for a massage or to buy dodgy CD's or sunglasses. Lo Pagan is good for nightlife and is only a short drive south (Spanish nightlife does not kick off until at least midnight). Benijofar (inland, north) is also worth a visit; try the Rocio Palace for professional Flamenco dancing.
All the urbanizations have their own bars, restaurants and places of entertainment. Some are good, but as a general rule they are avoided by the locals, especially the Spanish, as they do not represent good value for money. The only time I have seen optics used in a bar in Spain was in an urbanization and in one square in Villamartin they actually get away with charging 25 Euro for fish and chips!.
The web link that you posted earlier shows the Torre del Moro in the top right hand corner. This is in Torrelamata and surprisingly I can see it from my window as I type this.
I feel sure that the people that you are renting from can advise you further, although that is not always the case. Unfortunately there are many owners here that know little about the area except for the drive from the airport!
Here are a few more links that may help you:-
http://www.torrevieja.com
http://www.turismedelx.com/en/
http://www.elitepm.org.uk/rojales_alicante_Torrevieja_info.php
Have a lovely holiday!