I wrote about this topic many times before, and in short: NO, the fall of Constatinople most definitely was NOT direct cause for start of Age of Exploration.
Most of the premises for this thesis are actually incorrect. Most importantly, Portuguese started exploring Africa before 1453 which alone answers your question, but also it is probably important to note that Constantinople was not really that important for trade with Asia as much as Egypt and Levant, which were not under Ottoman control but Mamluk, and where trade was going uninterrupted. If anything Iberians just wanted to find Asia to get around all the middlemen, and buy directly and cheaply at the source.
Now, let's move to the second question. There were few voyages circumnavigating Africa in ancient times, most famous the one from Hanno of Carthage. That voyage, and some additional ones are mentioned in ancient works, e.g. the one I know of is Pliny's Natural History, book II, chapter 67 where they are retold. Pliny's work and specifically those voyages are referenced by some Portuguese of 1500s, meaning they were familiar with them. Unfortunately ancient times are not my specialty, so I can't go into details of those expeditions
Let's now switch back to Christian Middle Ages, which I am familiar with, and which I assume you inquire about. Before the Portuguese start of exploring Africa (usually given as 1415 when they took the city of Ceuta on North African shore) there was a famous attempt by two Genoese brothers, Vandino and Ugolino Vivaldi, who already in 1291 took a galley and sailed down Africa attempting to reach Asia. Nothing was heard from there ever again, but we do find some references to the attempt in Portuguese works much later, meaning in it somehow remained in popular imagination.
The next attempts I know of were Portuguese attempts down the coast of Africa. They were technically preceded by Castillian-French attempts to colonize Canaries in 1402, but overall not much exploration was done. So after 1415, Portuguese, under command of prince Henrique or Anrique or most famously known as Henry the Navigator, started sailing down Africa. They passed the southernmost reached position of Cape Bojador in 1434, after which they were systematically "exploring new lands". The exploration voyages were in the beginning actually more like slave raids, but effort to chart, note, and explore new areas was always made. By the time of 1453, they already reached Sub-Saharan Africa and rivers of Senegal and Gambia. Now technically, they didn't really want to reach Asia just yet (despite some ambiguous notes saying they had that general idea) and were mostly content with trade with Africa (or Guinea as they called it) but Age of Exploration had definitely started and those early voyages formed a foundation on which the exploration would continue until their famous peak in 1492 and 1497.
Now, were those attempts "state-sponsored"? Vivaldi brothers definitely not, they were a single merchants idea and execution. Was the Portuguese effort "state"? That's more complicated than it seems. Initially, prince Henry the Navigator was investing in the exploration, and he was Royal Family and was definitely backed by the state in some ways. He was granted monopoly on new discovered lands and Crown transferred some rights like taxation to him and his successors. However at the same time, Henry financed his own expeditions, investing his own money, money of Order of Christ he was in charge of, as well as money he borrowed from various foreign and domestics bankers and rich men. In that way, the expeditions were more a private venture, but heavily interconnected with the State.
After Henry's death in 1460, and you see we are already in post 1453 time, the Portuguese slightly stopped with exploration efforts, but revived it in 1470s in a nice example of state-private cooperation. The Crown leased the rights for exploitation of the new lands for 5 years for a monetary compensation and under a condition that 100 leagues of new coast is explored. This arrangement, very succesful, lasted until 1476, after which Prince João, future king João II, took full control of exploration back into Crown's control and with which it became a fully state-sponsored venture, and would remain so for some time.
