My better half should really be the one throwing her 2 cents in but she is at work. She is a lab tech for an invasive species lab with IFAS at UF. As for your multifloral rose problem cut the stem about 4 or 5 inches above ground level and use a brush on herbicide to coat the exposed portion. I don't belive Round-up will work for this aplication, but I am not sure.
Man, oh man, is Florida over run with invasives. Every where you look, and for a state that claims to be aware of the problem and trying to fix it they just keep making the same stupid mistakes. Here in Gainesville, a campain was lead that finally convinced the city to stop planting Chinese Tallow and remove the existing trees. Unfortantly the city replace them with Chinese (Paper Bark) Elms.
One of the projects that my grilfriend works on is a massive Ruellia study funded by the DEP. Ruellia brittoniana is already a major invader in Florida's wetlands, but you can go buy it at just about every nursury in the state and it is one of the most common landscape plants used.
And you know most people have know idea either. I saw somebody posting air potatoes for SASE (I can't rember where), I saw another guy in Florida posting Salvinia for sale. I have even been in so called native plant nurseries around here and found non-native species being sold as natives. People generally have no clue what they are doing when it comes to this stuff.
It is not just plants either, we have some nasty invasive bugs and animals. Florida hosts 23 species of Anole, 22 of those are invasive. We have established populations of at least 3 species of Monkey. Tropical snakes, all sorts of non-native fish, turtels, forgs, and the list can go on and on.
And now because of California's little mistake with the 2000 or so Camelias that got sent throughout the US, Florida is closed to plants from California. You known the Withering Oak Desease prolem they have out there? That is the real invasive that scares me. Just image the Southeast with NO OAKS. Man its a scary thought. Its hard enough dealing with what has happened to the Chestnut and the high elevation spruce/fir forest. Lose the oaks too and you have major ecological armageddon in the Appalachians.
On a happier note I will close with something a very strange friend of mine once told me (though I can't figure out why): "If I [he] were ever to become a dendraphile, tree-of-heaven my tree of choice... such soft bark" LOL. We are still not sure if he was kidding or not, but every time I talk invasives I think about that infamous statement.
P.S. Keep in mind I am a botanist, our sense of humor is very off.