Will your cheek really sink in after tooth extractions?

It is possible. The presence and/or absence of teeth can definitely have an influence on the support of the lips and lower facial third. An individual who has all of his or her teeth removed without replacement can and will likely have the appearance of loss of fullness of the cheek and lips.
Not really. Extractions without tooth replacement is usually what can lead to changes in facial appearance. This can be due to teeth shifting positions or "bite collapse" (where you chin is closer to your nose when you close your mouth, also known as a" loss of vertical dimension" in the mouth) this can be prevented with timely replacement of the missing teeth before damage is done.
They can. Cheeks can lose their support from teeth if they're lost.

Related Questions

What do I do? I have this weird feeling in my cheek after tooth extraction.?

See ur surgeon. Can't tell you why you have a 'weird" feeling.... Could be a pulled muscle, could be inflammation from the injection, could be something else. It needs to be physically evaluated, and you are under the care of a dentist... Give them a call for a follow up. Read more...
Nothing to be alert. Pain, swelling and difficulty in opening the mouth for a few days after a tooth extraction is normal. The weird feeling is a cheek swelling due to the tissue trauma caused by the extraction process. Recommended: 24 to 48 hours after the extraction – leave the swelling alone and do not apply any hot or cold compress. Read more...

Had dry socket treated 3 weeks ago after tooth extraction. The socket still hasn't closed and there's swelling on the side of gums near cheek still.

See below. Go back to the dentist that performed the extraction. Make sure to refrain from smoking to not deter the healing process and speak with the dentist about replacing the missing tooth. Read more...
Seek evaluation . Healing varies and is dependent on various factors including body's capacity to heal which is usually better at younger age and healthy body. The extent of surgery also affects the healing process. It is possible that there is residual tooth, bone fragment, or foreign objects such as food debris in the socket. Have it evaluated by the treating surgeon. Read more...