do plants and other animals depend on mankind for their existance?

Not all plants or animals perhaps but certainly some, most notably those we have bred or domesticated for our own uses. To give some examples:-

Many species of hybrid flowers and plants do not "breed true" if allowed to pollinated naturally (F1 Hybrids). They must therefore be artificially cultivated - by humans

Cows have been bred to produce more milk than their calves could possibly drink. This means that their udders would become so swollen with milk as to cause injury and death. Domestic dairy cattle depend on humans to survive.

Bulldogs have been bred with shoulders so wide that the mother is unable to give birth naturally - requiring cesarean section (by humans) for their survival.

Of course life would continue without humans but removing any one species, including humans, is inevitably going to have an impact on others

Eh, not really. There was both animal and plant life before mankind. If anything, they might prosper more, since as a species we are quite the invader. For example botht he dodo bird and the tasmanian tigers were eradicated by Humans.

I think both plants and animals would do very well without us. No animal has ever extinct another. It's rather the other way around, we depend on them.