Are you saying the files are disappearing, or the download information.

Are you saying the download icon is missing?

Type about:customizing<enter> in the address bar. In the new window, look for the icon. When you find it, hold down the left button on it, and move it to where you want it to be. You can move any of the icons you want from here, but; some icons are locked in place.

There is also a Restore Defaults button at the bottom middle.

Note: about:customizing is no longer present in Firefox 47+ releases.

• Press Alt or F10 to bring up the toolbar.
  Then View > Toolbars > Customize.
• Right-click on a blank spot in the toolbar and
  select Customize.

Do you see the downloads in the Downloads Manager?

Make sure you do not run Firefox in permanent Private Browsing mode (Always use Private Browsing mode; Never Remember History).

• https://support.mozilla.org/en-US/kb/Private+Browsing

• Options/Preferences -> Privacy -> Firefox will: "Use custom settings for history"
• Deselect: [ ] "Always use Private Browsing mode"

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Browsing & Download History

Open the Customize window and set what toolbars (Show/Hide Toolbars) and toolbar items to display.

• Right-click empty toolbar area -> Customize
• "3-bar" menu button -> Customize
• View -> Toolbars
  You can tap the Alt key or press F10 to show the Menu Bar

• if missing items are in the Customize palette then drag them back from the Customize window to the toolbar
• if you do not see an item on a toolbar and in the Customize palette then click the Restore Defaults button in the Customize palette window to restore the default toolbar set.