No, you cannot buy tiles outside the city radius of 3 tiles. You can use a great Artist to culture bomb those tiles or a Great General to build a citadel next to those tiles if you need the strategic/luxury resources there, but don't want to build a city.
The city will continue to expand through culture up to a maximum radius of 5 tiles. But you need a lot of culture for that, the Angkor Wat wonder in that city is very useful for that.
You can never work tiles outside the 3 tiles radius, but you can get the resources on those tiles.
Best Answer
The answer is YES. Based on my experience, AI's are buying units specially when they are surrounded(usually the Capital) or about to lose a war.
There are instances that an aggressive civilizations are buying units when they declare a war against other civilization.