James Reeves wrote a lot of poems for children. I haven't had the time to search for the whole poem, but this looks like fun poetry for young children in the style of "The cow jumped over the moon".
There are fantasy figures in this poem, first Longears, perhaps an animal with long ears such as a hare or a rabbit, then Tapperbill, one can only guess who or what that figure is. "to tap" can mean to hit lightly and as the word bark follows it might be a bird, perhaps a woodpecker. Painted-wings is a third figure, might be a flying insect, perhaps a butterfly or a dragonfly.
If you have problems with a poem you should first find some information about the author to see what kind of poems he writes. If you google for James Reeves you'll get a lot of information.Then you should see some of his poems to get an idea about his topics or the genre of his poems. And when you have questions as to a poem you should give a link to the whole poem and not only indicate three or four lines. From a fragment of a poem it is difficult to give reasonable information.
If I had to say something about the proper names of the animals in this poem I would say they are "speaking names", that is proper names that refer to a typical characteristic of a person or in our case here of animals.
The poem with the line "The cow jumped over the moon" has the first line "Hey diddle diddle", and en.wikipedia has some information about this funny nursery rhyme from about 1700 under "Hey diddle diddle".