If you placed the Time Machine icon in the Dock, and then moved that icon too the Trash, you have done nothing critical.
Icons in the Dock are specialized aliases - they represent the items they depict, but are not the items themselves. Any icon on the Dock can be removed (usually by just dragging it off of the Dock) or otherwise trashed, and nothing is done to the original item.
To get the 'control box' for Time Machine, try going to System Preferences > Time Machine.
If you would like to have a Time Machine icon in the Dock, open the Applications folder (click its icon in the Dock); if it opens as an array rather than as a regular folder, click the button named Open in Finder in that array. Then just drag the icon for Time Machine to the Dock, someplace between Finder and the dashed line; wait for the icons there to move aside, then drop it in.