1. If you want to go through the entire series, start with Fallout 1, then work your way through them all. The old ones still stand as good games, but are PC only.
Other than that, start with Fallout 3. (New Vegas is a bit more difficult and complex)
Like TES, the stories aren't connected. They just exist in the same world.
2. Yes and no. It's an open world, with lots of quests and RPG elements. But there's guns and technology, and the game is much more gritty and dark.
3. Yes. Lots of violence, advlt language, and advlt themes. But there's humour too.
4. It's mostly a first-person-shooter adventure, although you can play in third person. But that doesn't mean it's anything like Halo or CoD. You explore, do quests, kill everything that moves... so in that way it's more like Skyrim.
Character progression is different than TES games. You gain experience for completing tasks/quests, and when you level-up you can choose which skills to increase, as well as perks that give you various new talents.
Check some Youtube videos of gameplay to see how the combat works. It can be fast and crazy, but you can also use VATS to freeze time and attack strategically.