I know Egypt provided grain to the Romans and Byzantines, so it still had a surplus of agricultural products, but was the architecture, farming techniques, building techniques, city planning etc. there some of the best in the Mediterranean still? Or did they fall behind, and Greek and Arab scientists/engineers/artists/governors were responsible for its transformation into a new sophisticated center in the Arab era?
As a follow up, how much did the ancient Egyptians "teach" or "transmit" to the others?
I'd like to preface my answer by saying that this is a very complicated question, and there are archeologists, anthropologists and egyptologists who spend their entire career trying to fully answer it. With that having been said, I'll try my best to make it easy to understand.
To begin, we must get into the nature of your question itself by examining which periods of Egyptian history it calls to. We begin with general timeline of Egyptian history: Prehistory Era (Paleolithic to Protodynastic) - Dynastic Era (Dynasty 1 to Dynasty 30) - Greco Roman Era (Alexander to Late Antiquity) - Arab Conquest (Arab Egypt to Mamluks) - Ottoman Era - French Egypt - Muhammad Ali Pasha - British Egypt - Republic - Revolution.
Since the question is about Egypt following conquest by the Assyrians, Persians, Greeks and Romans, it spans the tail end of the Dynastic Era (Assyrians and Persians) and the Greco Roman Era.
Since the question concerns the status of Egypt (powerful, advanced, backwater), it would be helpful to take a closer look at Egyptian History during the Dynastic Era.
A common misconception is that the Pharaohs were incredibly powerful rulers and that the Dynastic Era of Egyptian history could be considered as the “glory days” of Egypt, the truth is that during the 3,500 year span of this time period, power and greatness were periodic to say the most.
Historians divide this period into the following pieces: Early Dynastic Era - Old Kingdom - First Intermediate Period - Middle Kingdom - Second Intermediate Period - New Kingdom - Third Intermediate Period - Late Period.
Did you notice those intermediate periods? If Dynastic Egyptian history was a TV program, those would be the filler episodes. The Intermediate Periods were times during which Egypt was made weak due to decentralization. This caused Egypt to erupted into civil war, the Pharaohs to be stripped of their power by the regional nobles and even (most of) Egypt to be conquered by foreign powers.
The main idea behind this, is that Egyptian greatness and advancement are directly tied to centralization of power and whenever that happens Egypt has bursts of glory. The most notable of these were the Old Kingdom (The Golden or Pyramid Age), the Middle Kingdom (The Classical Age) and the New Kingdom (Age of Empire).
Bringing this back to the question, did Egypt become a backwater after it was conquered by the Assyrians, Persians, Greeks and Romans? The answer is yes, but why that happened is equally important.
The simple answer is, a united Egypt is too powerful. Egypt is situated in one of the fertile, highly populated and economically productive regions in the world. And so as to prevent an Egypt from rebelling, its respective conquerers intentionally decentralized it after conquest, hence artificially keeping Egypt and Egyptians in the “backwaters” so as to make it easier to rule.
TLDR After being conquered, Egypt was intentionally kept weak and decentralized to make it easier to rule
References The Middle Kingdom Renaissance from The Oxford History of Ancient Egypt, The World of Ancient Times, A History of Ancient Egypt