1) Does training still occur during the offseason? YES
2) How do you tell when a week ends or begins?
The schedule is the first thing you see when you read the FAQ:
http://www.grid-iron.org/index.php?page=home&subpage=help
3) How do you tell when a season ends or begins?
Today was the last league game. I believe there are 9... or is it 10 league games. Just count how many you've had in the matches tab.
4) How do you tell who reaches the playoffs?
There are only cups (European thing) and not play-offs. Cups can also be read about in the FAQ.