How can one respond to Eid greetings from people who insult and mock the religion, and is it permissible to remain with them when they are not uttering such things, and is it permissible to pray for their recovery and well-being?

Reviling the religion of Islam is major disbelief (kufr akbar), while reviling the religion of a specific person is not absolute disbelief; rather, it may or may not be disbelief. Similarly, mockery (istihza'): if it is directed at what is known to be from the religion of Allah, then it is disbelief. But if the mocker does not know that it is from Allah's law, then it is not disbelief. Reviling the religion or mocking it does not authorize the general populace to rule that the doer is a disbeliever; rather, the determination of this rests with the Sharia judiciary or the mujtahid scholars. It is permissible to supplicate for worldly benefits for an original disbeliever and to return his greeting, so this applies even more so to the one the question refers to.