im an amature photographer and have taken some pet pics in my life. its obviously harder with puppies but a good trick is to tire them out first! once they are tired, they are a little more willing to sit still.
to get rid of the "green eye" all you have to do is not use a flash. most peoples houses arent light enough to take a good picture without a flash so what i do it "steal" a few lamps from around the house and flood the area with light that im taking a picture in. with small dogs like yorkies, one extra lamp will likely supply enough light if you are using digital. for film cameras, you will likely need a lot more light.
persistence is definitely the key though! there is no animal photographer out there that will get a good shot every time. take tons of pictures and you will likely get 2 or 3 that will be great!
happy photographing
(a couple examples of my photoshoot with my chldhood dog charlene (rip precious girl!) i used one lamp and overhead lights)