Yet mean "up to a specified time or up to present time" It's close as "still" Morning is interval time between after midnight and noon, moderately is 5 pm to 9 pm. So, Is it morning yet? means someone asking fore sure that this time (when she/he ask) it's still in the morning not noon.

Has morning come you could think of the first one. About the school work if you say not yet just means it isn't done.

The first "yet" means "already" based on how the sentence was used. "Is it morning already?" :)