Is working in the design of homes and mosques considered a perpetual charity (Sadaqah Jariyah) if money is received from the project owner, and can it become so by requesting a price lower than the cost?

There is no objection to the work mentioned. Working in the construction of mosques is not considered charity from you, but rather charity from whoever built the mosque and paid the wages and the cost of the equipment. Being lenient in pricing is rewarded, for the Prophet, peace and blessings be upon him, said: "May Allah have mercy upon a servant who is lenient when he sells, lenient when he buys, and lenient when he demands." You can share in the reward with the mosque's owner by donating your work or a portion of your wages, so that it becomes an ongoing charity (sadaqah (definition) jariyah).