What is the ruling on one who abandons prayer due to the difficulty and cost of purification and the impurity of most areas of the house?

Abandoning prayer is a grave sin; in fact, it is greater than adultery, theft, drinking alcohol, and taking a life, by consensus. It is considered disbelief by many scholars. There is no excuse for anyone to abandon it. What you mentioned about impurity appears to stem from waswasa (whispers/obsessive thoughts). Even if impurity were widespread in a place, it is sufficient to place a pure mat to pray on. Purity is a simple matter; it is enough to wash the limbs required for purity once, and increasing the washes in ghusl (definition) (ritual bath) or wudu (ablution) is from waswasa. Maintain your prayers and beware of abandoning them.