Tray is such an ambiguous word 
. We have notification area which is, well, a Gnome 2 tray. You can add or remove it from panel. Some applications still use it. For example, if you use several layouts, keyboard indicator will appear in notification area. But the majority of system indicators are displayed in indicator applet now. This approach was introduced in Unity desktop for several reasons and now is deliberately implemented in Ubuntu MATE. Vanilla MATE desktop, however, does not provide indicator applet area, so you can get a more gnome-2-ish experience in other MATE distributions out of the box.
So yes, in theory, you may remove indicator applet, add some standalone applets and force your apps to show icons in notification area. But Ubuntu MATE developers are doing the opposite .
Concerning tweaking the indicator applet itself:
- here is a nice discussion about spacing;
- to completely get rid of the clock you'll probably have to uninstall
indicator-datetime; - I am sure I saw some discussions about icon rearrangements inside indicator applet in this forum too.
Good luck!