An update on my situation.
I went ahead and updated my various security software of Spybot, Advanced System Care, and AdAware and went through my computer deeply. No issues.
Next up, I clicked on the Repair button on the top right of the client window and did a deep scan of my game files. All were confirmed as up to date.
Status unchanged. "Server Undefined".
I can play every single other game installed on my computer in both Offline and Online modes, ranging from Duel of the Plainswalkers 2013, Skyrim, Dungeon Defenders, R.U.S.E., City of Heroes, and many more.
No issues. With any of them.
Edit - Used the "thereisnourflevel" code with the client patcher. This did indeed let me bypass to the game splash page with my Account Name and Password entry. Entered the game, but no news would load up. I could call up my summoner info page (runes / masteries) as well as the game store. Attempted to play a match. Game gave me an error before reaching the Champion Load screen for the teams, with the error being Firewall related - blocking the client.
Again, I have Windows Firewall OFF. Never used it. Ever. I only use the passive defenses of Advanced System Care, Spybot, and Adblock Plus / NoScript plugins for Firefox browser. I have ALWAYS been able to play the game, and again, this has only started since Sunday.