You can install a plugin. On the Wordpress website you can find the one you like.

Chat room plugins you can find here:

http://wordpress.org/extend/plugins/tags/chatroom

Other interesting plugins can be found here:

http://www.wordpressfamily.com/category/plugins/

There are several plugins that allow you to add an event calendar to your self-hosted WordPress blog. Some plugins requires you to create a separate post for each even you want displayed on your event calendar others imports an iCal file and generates an event calendar. These plugins can be found in the WordPress plugin repository. Some example plugins (no endorsement implied) include Events Calendar, WordPress Event Calendar, and ICS Calendar.

No there is no automated builder as such. Plugins are developed by coders.

go to you dashboard/click on plugins down on the left hand side/your now at the plugins directory/if you want to look for plugins to add to your existing plugins/click on add new and search.

There are a good number of widgets that can be found when searching the official WordPress plugins site. Also, many plugins contain their own widgets, which again you can find on the WordPress plugin site. You can also write your own widgets for use in WordPress and include the code in your own plugin or inside your themes function.php file.