How exactly are Kingdom cards supposed to be played? Are they a card all players share and/or do all the effects on the card apply to all players? I noticed some cards mention power on Kingdoms counting towards your victory total. What does this mean?

Sorry if I missed an explanation somewhere, but I searched for the term "kingdom" on this forum and couldn't find any mention of this particular card type. I just feel like I missing something with these cards.

"Kingdom" is just a trait. It does nothing on its own. The only time it would come into play is if some other card references it.

So location cards that don't have the usual parchment header that say "Kingdom" on them are played like normal locations? Why do they look different than other locations? Are they played on your side of the table like usual? Or once they come out, are they in play for everyone (since their text seems to reference all players)? What are some strategies for playing these particular cards that seem to benefit all players.

They are just standard locations with the "Kingdom" trait. You follow their text like any other location. If it is worded in such a way that it could benefit any "player," it does. If it is worded in such a way that only the controller benefits, it does. For example, Bay of Ice lets the player who wins initiative draw a card (no matter who controls the location) while other Kingdoms have triggered effects - meaning that only the person controlling it can trigger the effect.

The unusual graphic design comes from the CCG days (when they were first introduced) and they were more likely to be double-edged swords.