I usually use the midland style because that's the style I was originally taught by Nigel Adams and because it's a beautiful style, if somewhat wasteful with the binders used on top. It makes for for a very dense and relatively strong hedge.
That being said there's a lot of other styles each with their own histories and use cases.
If you want to learn there's some books on the topic, though not all of them in English. For instance the Dutch stichting heg & landschap has a decent guide and overview of the most common styles and techniques in the Netherlands and Flanders (Heggenvlechten en haagleiden in Nederland en Vlaanderen). A very in depth one is "Europe's field boundaries" by Georg Müller, but I suggest trying to find it in a library as it's very expensive.
In order to actually learn the techniques the best way is to find a teacher or course near you. There's a lot of videos on youtube and pictures in the aforementioned books, but those aren't really a replacement for someone experienced showing you the ropes.
I'm not exactly sure. A chain link fence is a one time expense as opposed to a hedge which is a living, growing thing and so needs continuing upkeep. And yes, it can take a couple of years before the hedge is ready to be laid... There's also the used space to consider, as a hedge is a lot wider than a fence.
I guess it really depends on your specific situation.