Welcome to the forums miccro.
With regard to buses, you are correct about the number 20 being a very limited service. However, it is not the best service for Altea and Albir, you should be looking at the number 10 which runs at 15 minute intervals through the day and at 20 minute intervals through the evening, up until around 11pm.
There are bus stops on the main road close to Gemelos 22. However, despite the frequency of the service, buses from the Old Town end of the resort can often be full by the time the reach that point, especially just after hotel meal times and on market days when tourists are on the move in greater numbers. But avoiding peak times, you should have no problem boarding the bus close to your apartments.
I believe the adult fare is ÃƒÂ¢Ã¢â‚¬Å¡Ã‚Â¬1.20 single.
I will leave others to tell you about evening highlights and places to see in each town.