Do these kinds of frogs really exist?

Look at the little, fat frog at the very bottom of this old German illustration. Also, the weird frog with the huge webbed feet that looks like it’s flying through the air.

Are those frogs real, or was Ernst Haeckel licking too many toads?

Yes, there’s nothing in that picture that is utterly absurd in terms of amphibian morphology. Certainly the illustrations are a little stylised in their presentation, but yes, there are tree frogs that glide on outstretched webbed feet and there are fat,globular frogs.

The rounded one looks almost like a stylized Pac Man frog. Yup. Just like the video game.

Saw one that looked just like that yesterday on a Discovery Channel program about the Rufiji River in Tanzania. It hibernated underground most of the year and looked more like a half-deflated football with an owl’s face than a frog.

Here’s your flying frog:

http://www.ecologyasia.com/images-k-z/wallaces-flying-frog.jpg

Just beware of this guy: Mexican Staring Frog

Yes, although somewhat stylized all the frogs illustrated are quite real - the caption gives the current scientific name. The small fat one you mention is Breviceps mossambicus. Laughing Lagomorph has already supplied the link for the Bornean Flying Frog Rhacophorus pardalis.

Perhaps even weirder is the Mexican Burrowing Toad Rhinophrynus dorsalis

He looks like he’s been detailed.

“Don’t like your plain green frog, bring him to Maaco!”