I recently got addicted to a game called Alcazar on Android. The premise is simple, you draw one continuous path inside the player area, pass through every square once, then out the player area. It was very simple to learn but had a good depth of strategy. It was one of the best puzzle games I have played. I would recommend taking a look at it.

1. Easy to learn the basics but it should offer interesting gameplay.
The puzzle should be decomposable meaning you can solve the puzzle in parts. Solving one part gives you clues to solve another part. Think of a Sudoku puzzle. Solving one 3x3 square gives you information to solve another 3x3 square. The parts are interconnected. This lets you work in a part, if you get stuck you look at other pieces.

2. I don't know about trends but for my personal taste I don't like puzzles that mostly take a simple strategy and brute force. I should be experiencing "aha" moments while solving the puzzle.

3. Just a good core puzzle mechanic with enough diversity to not get bland. I don't need fancy visuals, although its hard to market a game that doesn't look good.

4. Not that often. I usually play when I have a few minutes where I need to wait, so I can't commit to long puzzle session.s

5. Paid in advance. Alcazar was free to download but you had to pay a one time price to unlock all the puzzles.

6. Not sure, I don't play much.

7. I like to concentrate and do some deep thinking.

I assume you are talking about something like Candy Crush, not adventure games?