With at least the lightdm-gtk2-greeter; even though it visually has focus upon startup, the password entry box eats the first key GAINING focus when the mouse pointer was on it (or on the user-select box).
Both the login window and the mouse pointer are centered on startup, and for me and with at least the gtk2 greeter, this means that the mouse pointer is actually on the password entry box -- and therefore that lightdm seems to malfunction. It's only until I figured out that I first needed to kick my mouse out of the way upon startup that I understood that lightdm wasn't in fact more seriously broken...
I've "fixed" this manually by specifying a position= statement in the config file that avoid the centered mouse being on the password entry box after which things work fine, but this is of course not a nice thing to have to do.
Although I don't suppose it's THAT specific: I'm on a 1280x1024 screen...
I can confirm this bug. When tooltip is shown it receives focus and "eats" first key.
Gtk3 version is affected, it's just not too noticeable most of the time.
1. Click on password entry (focus it)
2. Move cursor to any widget with tooltip (users combobox)
3. Focus moved to tooltip - press any key, it is will be "eaten"
Fixing must be simple (If I don't missed something), I will upload patch soon.