Civilization – What rule does border expansion follow

No.

It's my experiance that simply having a tile that contains a resource within your borders and with the relevant improvement constructed (and not pillaged) is sufficent to add it to your network (it doesn't even need a road/rail connection like they did in Civ4) - this is for luxuries and strategic resources, things like sheep that just boost tile output obviously have to be worked (in the same way as you have to work a hill to get the benefit of having built a mine there).

And in a recent game I culture bombed my territory outwards in to a neutral desert area, specifically just to obtain an oil field. It was 5 tiles from my nearest city, and a soon as my worker finished building a well upon the tile I had oil available to be used for construction.

The enemy cannot block your access to the resource (say by standing units on that tile) directly, but it is quite a simple task for them to pillage the square in question, which will then cause you to lose the resource until such time as you repair the pillaged improvement.
(With thanks to bwarner for the clarification in comments).

Note that most (all?) resources also grant an output bonus to the tile they are on, and you have to work the tile from a city to get that bonus.

I can confirm a very good effort for a science win from AI. They have built an Apollo Program and several ship parts. It was a really nice end-game race, as I was beelining tech tree to rush United Nations for diplomatic. It was my first successful immortal game so I didn't want to lose it just to see if AI can win, but it certainly looked like they could.

On the other hand I had another game where I was hopelessly lagging behind a runaway AI on science. They entered future era, when I was still in mid-industrial. But they were so busy conquering the world that didn't bother to go for science victory. So it might be that if AI is set up to achieve dominance they won't go for other types if victory.