This time I did
apt-get remove lightdm-gtk-greeter lightdm-gtk-greeter-settingsso that it won't conflict with slick-greeter.
You shouldn't worry too much about conflicts. LightDM itself is an engine that manages the login screen. It doesn't work by itself but needs what it calls greeters, which are a bit like plugins and use different toolkits (e.g. there's one that uses GTK, another that uses QT, there's one that uses WebKit −a HTML renderer used in web browsers− and so on; Slick uses GTK too.) They offer different methods of customizing the appearance of the login screen for themes creators.
You can have several greeters installed, they won't conflict, the one used is defined in the LightDM config (in a peculiar way IIRC).
The wallpaper used is set in the LightDM config too. You can easily change various options using a GUI tool as you have found out with the one that is in the Control Center. A small sidenote here: you'll notice that there's an option to use the user's wallpaper on the login screen. If you enable it, you'll see that it doesn't work. This is a bug in Ubuntu MATE. I have a script that fixes this in a kind of hackish way here though it would be better if the bug is properly fixed one day in MATE.