To your second question, while I can't speak to the earlier versions of it that took place several hundred years earlier, the modern investiture ceremony was designed to be a pageant and was initially suggested by one of Queen Victoria's daughters years earlier, ignored by her brother Edward VII when it came to the future George V, and then shelved until Lloyd George became Chancellor of the Exchequer in 1908.

As it turned out, after Edward VII died in 1910, Lloyd George was conveniently the MP for none other than Caernarfon Boroughs and was the Constable of the Castle. There were political motivations for both the Crown and Lloyd George in this. The former needed to solidify his position a bit as his father had barely kept a constitutional crisis between the Lords and Commons under control earlier that year; the latter both wanted to deliver something to his constituency (there were a lot of Welsh Dragon flags to help make the point) as well as prove that he wasn't quite the radical that many felt after he played a significant role in creating said crisis over Ireland and Home Rule.

The planning and execution of all this was the responsibility of the Home Secretary, one Winston Churchill, who never turned down an opportunity for formality and pageantry throughout his life. He tutored the future Edward VIII quite a bit, had a role in the ceremony proper, and grew quite close to the young lad even if it wasn't fully reciprocated. As I've written before, later this was part of the reason why he disastrously stuck by his side during the abdication crisis.

To quote from Ziegler's King Edward VIII:

"Some time-honoured traditions were hurriedly invented, Caernarvon Castle refurbished, gold quarried from the Merionethshire hills to make the Prince’s regalia, and a quaint costume of white satin breeches and purple velvet surcoat devised for the occasion. At this point Edward struck. What, he asked, ‘would my Navy friends say if they saw me in this preposterous rig?’ The Queen talked him into grudging acquiescence and Lloyd George taught him some Welsh phrases for the occasion."

So in short, the modern Prince of Wales has an investiture largely because two of the canniest politicians in British history thought it was good politics in 1910, and it's now tradition despite no legal requirement for it whatsoever.

William was not immediately Prince of Wales, the queen died September 8th, Charles became king and William became Duke of Cornwall (in England and Wales) it was only the next day that he was made Prince of Wales.

It’s not a title that is automatically inherited, the monarch gives the title to their heir at a time of their choosing. Charles became Prince of Wales in 1958, 6 years after his mother became queen. He was the first Prince of Wales since 1936 as neither his mother nor grandfather had held the title prior to becoming monarch. The investiture ceremony 11 years later was mainly a piece of theatre to mark his 21st birthday and his entry into public life as an adult.