Pheasants I usually use 12ga, 2 3/4", #6, high brass, 1 1/4oz, lead, Winchester. For long shots or a third shot I use 12ga, 3", #6, 1 7/8oz, lead, Federal, but a close bird with these is destined for the soup pot. They used to be 2oz loads but they changed it. 5's and 4's are also good but I like the extra pellets with #6. Modified choke almost always.
A friend had success with a 20ga on pheasants but his was a semi-auto and he tended to fire too fast and run out before downing the bird compared to how he did with his 12ga pump. He did the same thing with rabbits and that gun.
A 12ga 2 3/4 isn't really suitable for duck with steel other than for cripple loads. I wouldn't try a 20ga on ducks with steel. I don't know anyone who does, there must be a reason.
With Hevi-Shot, yes a 20ga would work, but at $2 a shot it's not worth it to me. It does a great job, it just costs too much. I don't know about a 20ga with Hevi-shot on geese, you may limit yourself out of some great meat by going with a 20ga if geese come in.
Ducks with steel depends greatly on what the gun likes. I couldn't get my old 500 to pattern any steel consistently. I got an 835 and it does much better.
My friend suggested BB's. I was hitting ducks but they would not go down. The ones we got were showing no body penetration. Thinking it was lack of pattern I tried #2's at another friend's recommendation. My 835 puts out huge and tight patterns with #2 so I know those ducks were getting hit everywhere. The results were the same as with the BB's, few ducks down and what did come down were cripples.
I was goose hunting with 3 1/2" BBB's. Some ducks came in, some ducks dropped. Same results on most of the ducks that have come in since I started using only 3 1/2" BBB's. I'm penetrating into the body cavity and having no problem breaking wings, a much higher percentage are dead on the water. The 835 patterns the lower pellet count tight enough that a duck in a 30" circle @35-40yds could not fly through it. It also has the energy to break goose bones. The ones I use are Winchester Drylok, 3 1/2", 1 9/16oz, BBB. Xtra full T-F choke. The thing I really like about this load working so good on duck is that it's equally good on geese, I don't have to shuck ammo while trying to call in geese now.
I'm not saying that you should use the 3 1/2" BBB. I'm saying you're going to have to get an assortment of chokes and an assortment of ammo and pattern your shotgun to see what's going to work. If you can pattern a light payload of #4-#2 steel it may go fast enough to penetrate, enough people have proven that. Most people would tell you that a 3 1/2" BBB wouldn't pattern enough to hit a duck and that an Xtra full choke would pattern so tight that you couldn't hit anything moving over decoys, I have proven otherwise.
Experiment and use what doesn't work good for cripple loads.
If it's worth it to you to use Hevi-shot by all means do it. That stuff is great. Use what worked with lead.