Ubuntu allows you to install several desktop environments all at the same time, and you can switch among them when you log in. Unfortunately, most are not "refined" to work together (as you have seen) so you see pieces and applications for other desktops scattered about when you have them installed.
(Removing the lubuntu-desktop package doesn't necessarily remove all of lxde; it just removes the package that brought in the parts considered essential to lubuntu. The same is true of removing ubuntu-desktop.)
You can TRY using the terminal command
sudo apt-get autoremove
and see if more of the environments you don't want to use get removed, but there will likely be some additional extra packages hanging around regardless.
One solution is to start from scratch and install a distro that uses Mate as its desktop and isn't cluttered with the items that aren't made for it. Another option is to just ignore the things that don't seem to work with Mate.