Working a tile does not seem to change the ownership of that tile. If the city is conquered, only the tiles the city directly acquired (culture or bought) are transferred to the conqueror.
Placing a city into an area already owned by the player will transfer ownership of the city tile and the initial inner ring of tiles to that new city.
Notice there's a difference between improving a tile and working a tile. Usually you will improve tiles around your cities and then send citizens to work these tiles, but this connection is not necessary.
Improving a tile
To improve a tile, you need it to be within your cultural borders (the tinted area) and reachable by a worker/work boat, and you need to have the appropriate technology. Once a tile is improved by the worker, you get any luxury or strategic resource on it, but that doesn't mean you get the tile benefits (e.g. +production, the small hammers) - you have to work a tile for that.
Pay attention: cultural borders can reach practically everywhere, since cities can expand these borders up to 5 tiles and great artists can expand it arbitrarily further. I once used 3 great artists to reach a far away tile (Aluminum...), it takes time because there's a cooldown between cultural bombs, but it's doable. "Culture bomb" is the name of the great artist's ability which does it. However, in practical terms, if you want access to any resource the best way is to build a city up to 3 tiles from it. Usually when I build a new city I only consider things in radius 3 to be interesting, anything beyond it just usually takes a lot of turns to get.
Notice that some worker actions, such as cutting down a forest or building a road, do not require the tile to be within your cultural borders.
Finally, if you build a city directly on a tile it doesn't count as improving the tile (the tile benefits won't change) but you do get whatever resource is on it.
And as Raven mentioned, to improve tiles at sea you must construct a work boat from a coastal city. Unlike workers, work boats are consumed when they construct an improvement, but they are otherwise very cheap.
Working a tile
To work a tile, you need it to be within 3 tiles of one of your cities, and then that city gets the benefits listed on the tile. In your screenshot, Krakatoa is unworkable, and no civilization will never get the benefits listed on the tile. Notice that even tiles in range 3 are not automatically worked, a citizen must be allocated to it. You can manually allocate citizens from the city screen but the computer takes care of it by itself, if you're new to the game you don't need to worry about it at this stage.
Your puppet cities also send their citizens to work nearby tiles, but you cannot change that allocation.
A tile doesn't need to be improved in order to work it, but most tiles benefit from improvement so ultimately it pays to improve any tile within range 3 of a city. Improving a tile beyond range 3 which doesn't have a resource on it offers no benefits (unless you improve it to a fort or something).
Best Answer
No, a tile may only be worked by a single city on any given turn.
If you tell a city to work a specific tile that another city is already using then the other city will be forced to stop working it. It is possible to get the other city to start working the tile again (but only by stopping the new city from doing so) - the tile is never "locked" to a particular city.
You should determine what bonus output the wonder has and then try to work it from a city that takes the most advantage of that - so if you have a gold-generating wonder, let a gold-generating city work the tile, for example.