Is it sufficient to return the greeting in writing or by moving the lips in emergency cases, such as a larynx infection, and does the same ruling apply to saying "Yarhamuk Allah" to one who sneezes and similar situations?

Returning the greeting of Salam is obligatory immediately. If the Salam is given to one person, returning it is an individual obligation (fard ayn) upon him. However, if the Salam is given to a group, it is a communal obligation (fard kifayah). If you receive a Salam via message, it is obligatory to respond in writing or verbally. The response must be audible to the one who gave the Salam. If a verbal response is difficult due to harm, one can respond with a gesture while moving the lips or with a faint sound. Regarding responding to a sneezer (tashmit al-'atis), there is a difference of opinion on its ruling. The more apparent view is that it is obligatory for whoever hears the sneezer praise Allah (say "alhamdulillah"), and the obligation is dropped when unable to do so.