Good to hear that you are making progress!
The apartment you mention by Viajes El Corte Ingles in Avenida de Rosalia sound like a good choice. My only concern would be street-noise since the street outside has a lot of traffic. That area of town is good (there is not really a bad area of Santiago). And you are right by the park, close to a supermercado (out the door of the travel agency, turn right, down the hill, turn first left... about a 100 meter walk for you).
Regarding furniture... hmmm.. thinking.... don't know of a place that rent furniture.... don't know of a 2nd hand market either...
If you know someone with a car you could try the Leroy Merlin Store on the way out to the airport,
http://www.leroymerlin.es. It is not really a furniture store, but they have a few shelves and stuff like that.
They have some cheap stuff in the super mercado at El Corte Ingles... and also in the supermercado at Area Central (located in the basement).
Neither of these are furniture stores, but might have some furniture pieces that does not cost too much. El Corte Ingles ships stuff home for you for a few euros more if you do not have a car.
If I think of another idea for a place for buying furniture I will post it here...
Good luck!
Ivar