Plants seem to sleep as well, can't pinpoint the exact source right now, but it was from a documentary. Something among the lines of "the hidden lives of plants"
I think it is possible for life to evolve on a planet with only plants and fungus.
That's kinda circular, given that plants and fungi are examples of evolved life :)
You are correct that plants don't strictly need help with pollination (they simply release their spores to the wind/current and hope they land on a receptive partner), however the odds of a succesful pollination decrease sharply as the number of competing plants in an area increase.
In my (uneducated) view, in a scenario where all life is stationary (rooted to the ground, hur hur), you are much more likely to find monocultures -- where most plants can interbreed -- or more species using asexual reproduction.
More importantly, plants evolved in the context of animals; if animals didn't exist, plants wouldn't have evolved seed dispersal or pollination strategies relying on them. They might have even evolved strategies we've never seen, that don't work well when there are animals running around eating everything.