This is Hothouse by <PERSON>, originally published abridged in the US as The Long Afternoon of Earth. Your description sounded familiar, but I read the book maybe 25 to 30 years ago, so the details were fuzzy. I found a copy at a local bookstore and all the points you remember are in the book.
In the book, Earth is tidally locked to the sun and the moon has drifted to one of the Earth-sun Trojan points. The side facing the sun is covered with a jungle formed by a single banyan tree. The story starts with a group of humans, led by a woman named <PERSON>, living in the middle layers of the jungle. The elders of the group "Go Up", which is a funeral rite where they climb to the jungle's canopy, and hitch rides on board Traversers, giant plants that have spun webbing between Earth and the moon. The story from there follows <PERSON>, the eldest male remaining, as the whole group gets carried away from the jungle. <PERSON> is banished from the group and has several adventures trying to get back to the jungle. Along the way, he encounters the morel, an intelligent fungus, and the Sodal Ye, the dolphin you remember.
* The size of the humans: when <PERSON> first meets the morel, the morel tells him:
"I know much about humans. Time has been terribly long on this world, and on the worlds in space. Once in a very distant time, before the sun was hot, your two-legged kind ruled this world. You were large beings then, five times as tall as you are now. 322
You shrank to meet new conditions, to survive in whatever way you could." (From Chapter 10)
* The metamorphosed humans are called flymen. They are described as:
In some respects they resembled humans. That is to say, they had one head, two long and powerful arms, stubby legs, and strong fingers on hands and feet. But instead of smooth green skin, they were covered in a glittering horny substance, here black, here pink. And large scaly wings resembling those of a vegbird grew from their wrists to their ankles. Their faces were sharp and clever. Their eyes glittered. (From Chapter 3)
* When <PERSON> and the other elders reach the moon, they have changed into flymen. They meet more flymen, who tell them:
"All who make the journey from the heavy world become changed. Some die. Most live and grow wings. Between the worlds are many strong rays, not seen or felt, which change our bodies." (From Chapter 5)
* The dolphin is called the Sodal Ye and is carried overland by a human:
A giant fish shape, much like one of the dolphins she had seen during their voyage over the wastes of the ocean, had been carried by a stooped old man. (From Chapter 23) 322