I'm not sure I believe the source you quote, since the Amshinover Rebbe is known for "running late" but not for skipping days. But here are some answers to your questions:
What shmoneh esrei does such a person daven Sunday morning
Most people daven on Motzai Shabbat before they have ended Shabbat, yet they say the weekday Amida.
One may actually daven Maariv on Motzai Shabbat before "3 stars come out"; some people do this regularly (e.g. "keep" Rabbeinu Tam, yet daven earlier), others do so on Chanuka so as to light as soon as possible. Either way, on Motzai Shabbay one always davens the weekday Amida.
So too on Sunday morning he would say the weekday Amida.
Does this person include retezei in bentching? Assuming he does, would
this person have to repeat bentching if it was omitted?
One is not allowed to eat after sunset on Shabbat afternoon before Havdala (Kitzur SA 96:4), so we must be talking about somebody who started eating on Shabbat. This person needs to say Retzei.
If he forgot, he wouldn't repeat the bentching , like anybody else who forgot Retzei at Seuda Shlishit. (Kitzur SA 44:13)
How does this not violate the prohibition of adding to a mitzvah?
Tosefet Shabbat is a Mitzva; so he's simply stretching an existing Mitzva.
Which day of the omer would he count Sunday evening?
By now you already know what I'm going to answer: He would count the correct count.
Reminder: One who forgot, couldn't or purposely didn't make Havdala on Motzai Shabbat, may do so until the end of Tuesday [afternoon]. (Kitzur SA 96:15)