It is interesting to speculate- I think he would have been a '' in charge'' ruler later, not like his grandfather perhaps, but even so young, he seems to have had a distinctive character, good or bad. There probably wouldn't have been a Catherine the Great had he lived, since she only came in because she married Peter III, who would have had a much happier life home in Holstein. Elizabeth wouldn't have reigned, but might have been happy enough with her lifestyle anyway, I don't think it was bad during his reign. I don't think Anna Ivanovna would have ruled either, certainly no great loss for Russia. It is hard to know what direction his reign would have propelled Russia into- but his court would have been a very decadent one, basically like that of Elizabeth or even Anna Ivavnova , however I believe he would have been VERY interested in ruling. But, these are speculations about a boy who died at about 15.
Yes, that's right. There never would have been a Catherine the Great - I wonder who she would have married and what sort of mark she would have made on history had Peter III not been available. Perhaps she might have married him anyway - he was her second cousin and already heir to the throne of Sweden. Her parents were angling for Catherine to marry him long before Elizabeth adopted him, and with Catherine at his side, perhaps Sweden would have regained some of its domination of Northern Europe.
I don't know how happy Elizabeth would have been if Peter had lived. When he first became Tsar the two of them were very close and he was said to be in love with her. However by the time he died he was engaged to Catherine Dolgorukova, and the Dolgoruvky family were doing their best to keep Elizabeth in the background. She might have had a happy life living on her estates - apparently she loved the countryside and the people who lived there - or she might have ended up marrying some German Prince.
As you said, Anna Ivanovna not being Empress wouldn't have been much of a loss for Russia. However, the one thing she did do was reestablish St Petersburg as the capital. Peter, who hated the place, complaining that he had no wish to live somewhere surrounded by water (or words to that effect) had moved the capital back to Moscow and showed little interest in European affairs or Russa's place in them. Having spent so many years in Courland, Anna had adapted to the European lifestyle and was keen that her court and subjects should consider themselves Europeans too.
One other consequence of Peter surviving is that poor Ivan VI would probably never have been born. In light of the terrible way his life worked out that might have been a blessing.