a unique area, being a while not by it, but probably will go close to it this summer,as my annual time in Spain is up.
the time of the year is good, unless you ventures into the mountains can be chilly.
to try to answer your question in my opinion, probably with a local guy taste is
1) best bet in Jaén, and you must check if still does, the info is about 5 yrs old: is Condestable Iranzo, city center, big rooms AC with parking garage, about 65 euros in your time frame. Next to the bus station is Rey Fernando with AC for about 55 euros.
2) the best in my opinion is Casa Antonio,AC with private parking (i always use the car) handle the food by Antonio himself, each dish carries a different plate design ,with good choice of wines: dishes go from 8 to 10 euros try salteado de chipirones al curri con verduras crujientes (== little squids salted in a tomato base sauce with curry and crunchy vegetables).
3) Do not know the paradores, never try them in the area
4) loaded with Romans ruins, and one of the best iberian museums in the provincial museum of bellas artes; thursdays is market day, Arco de San Lorenzo,cultural arts exhibits, the center is a historical monument. Capilla de San Andrés;Castle of Santa Catalina of arabics origins, Catedral amongst many sites .
very tranquil area, nothing bad to point out.
5) do not know about walking trails,but the weather is pleasant ,no blockage of passes at that time .
If you have time visit nearby towns such as Baeza,nearby is the torrèon Puerta de Ubeda,you can get good value products from the region including olive oil,and ceramics. Bailén is another nice picturesque town, first defeat of Napoleon troops in Spain.Cazorla, many museums and horse trails rides if still running you can contact Marcos Ruiz at +34 953 721 910.
Linares, where in its bullfighting ring die one of the great ones of all time,and my own favorite Manolete (manuel rodriguez sanchez), also the birthplace of one of Spain and the spanish speaking world best singer, the monster of the song, Raphael, El Niño de Linares (the son of Linares) also Ubeda, many museums of note the Hospital de Santiago (st james hospital) a smaller replica of El Escorial near Madrid.
enjoy Spain,just a proud ending.
