Duo! how many times have I told you that it's not OK to eat your friends!
Are we obliged to use "jest" here? Can it be omitted?
No, it can't, it wouldn't make any sense grammatically.
You can sometimes omit 'jest' in sentences like "This is X" (To jest duży pies vs To duży pies), but it's hard to say when it's perfectly okay and when it sounds strange.
Is grammatically correct to say "To moja smaczna kaczka"? Not in this case, I mean.
Yes, although it is a bit unusual. "This is my duck" is more likely to be said than "This is my tasty duck" ;)