Hi,
Well I've noticed that literally all the games on playfirst gives the opportunity for a 60 minute free trial, but on websites like Yahoo or Shockwave you sometimes only get offered 30 minutes.
Certain games such as Diner Dash gives you an unlimited amount of free game play up to level 6 or something like that!
The answer to your question of whether there is a online section is yes! If you click on games at the top of the website, then you will see on the left hand side, there is a little column. It says 'All platforms', 'Windows', 'Mac', 'Free online games' and 'Mobile'. Click on the free online games and then you get a wide selection of games. But be warned some of these games can be downloaded aswell, so once you get to a certain stage in the game they will say "If you would like to continue playing the game, download the 60 minute free trial", or something along those lines!
Hope I helped you!
 

I have to admit the amount of time managment games have decreased, but some have been relaesed. For example all of the dash games (diaper dash, cooking dash, diary dash etc). But also their is ice cream craze, Amelies cafe etc. You just have to click on games which looks interesting and it says whether it is a time managment game or not!
Hope I helped!