'Finding' is not possible. It's true house was already clean and tidy. But the action 'find' was not already the case. The mother *found* the house tidy after she returned home. 'To' here does not mean 'in order to'.

The simple past ‘and found’ is an alternative to using the infinitive. It depends on the context.

Both "to find" and "finding" can be used in the sentence, but they slightly change the nuance of the sentence. "My mom came home from work to find the house clean and tidy." In this version, the emphasis is on the purpose or reason for her coming home – she came home with the intention of discovering the house in a clean and tidy state. "My mom came home from work, finding the house clean and tidy." Here, the focus is more on the concurrent action of her arriving home and discovering the house in a clean and tidy state. It emphasizes the simultaneity of the actions. Both constructions are grammatically correct, and the choice depends on the nuance you want to convey. If you want to highlight the purpose or reason for her return, use "to find." If you want to emphasize the simultaneous actions, use "finding."