What is the ruling on wearing leather shoes that are impure on the inside, outside of prayer times, and is prayer performed in them valid?

There is no impediment to wearing ritually impure shoes outside of prayer when necessary. However, if there is no need to wear them, they should not be worn if they lead to the impurity of the foot, because deliberately smearing oneself with impurity is forbidden.

As for its effect on the validity of prayer: if the impurity transfers to a sweaty foot, then the foot must be washed before prayer. However, if the impurity does not transfer to the foot, then it does not affect the validity of the prayer. And if the impurity of the leather is due to it being the skin of a dead animal, then leathers are purified by tanning.