Would it be illegal for me to plant flowers in a public park?

Technically the parks are managed by the parks and recreation department. In reality no, plant away. Worst that will happen is they will pull them or mow over them when they grounds keeping.

Yes, it will be illegal. Vandalising public property. Odd to think of planting flowers as vandalism, but if the property doesn't belong to you...

That entirely depends on where you're at. There are probably no regulations on planting, but there may be regulations on digging. Realistically, plants plant things all the time.

Idiot get permission from the city first

technically..but look at this:

http://www.guerrillagardening.org/