How do I disable this feature of the Blizzard UI?

[Feldir]

How do I disable the default notifications for solar/lunar eclipses? Also, any recommendations for for replacing the frames I highlighted?

http://i.imgur.com/8OQbOHT.jpg

[Crudor]

How do I disable the default notifications for solar/lunar eclipses?...

Esc -> Interface -> Floating Combat Text -> untick Spell Alerts

[Feldir]

Thanks. Any advice for the latter?

[jmrice7]

For a Unit Frame change I recommend X-perl or tukui. Although Tukui is more of a complete UI change.

[ComputerNerd]

Thanks. Any advice for the latter?

There are literally dozens of options there.
Shadowed Unitframes, stUF just replacing the frames but still offering a lot of in-game customisation.
There are layouts created with oUF but those generally require some knowledge of the lua scripting language to modify, and so not recommended I feel for most users.