Not if you don't want oily jods and to slide off it! Saddle conditioner (containing beeswax and lanolin) is good for a new saddle. Put it on after you ride, then leave it to soak and buff it off before you ride.
No don't oil a saddle. The most you should use is a good leather feed - Stubben Hammanol. oiling leather is actually really bad for it - and oiling a saddle will cause future problems. If the oil gets in to any of the flocking it will just be soaked up and the flocking will mat together. Oil also stretches the natural fibres in leather which while it gives you the softeness short term actually shortens the life of leather as it goes too soft and looses it's natural flexibility. Stirrup leather, reins and girths should never be oiled as they are your life savers, if they are oiled they wll stretch - until they can stretch no more then they snap.
If your saddle is stiff then the more riding you do the softer it will become - if you want it soft before you ride simply roll the panels in your hands and they will soften up very quickly. The same rolling works for reins, bridle and stirrup leathers