Is it permissible to use a loan designated for purchasing home appliances for other purposes? And is it permissible to purchase from a seller other than the one from whom the invoice was issued?

It is not permissible for someone who has been loaned money for a specific purpose to use it for another purpose, because Muslims are bound by their conditions, and the general rule for conditions is validity and permissibility unless there is evidence to the contrary. These conditions must be fulfilled, based on the general meaning of Allah's Almighty saying: (O YOU WHO HAVE BELIEVED, FULFILL [YOUR] CONTRACTS). If the lender does not care about the purchase being from a specific place, then there is no harm in buying from another store. Otherwise, the condition must be fulfilled.