Okay, first rough bash at this is up there on the site, see the "Play Calendar" link in the Learn Submenu. Please have a play and fill up some slots so we can have a better idea whether this format will work okay. You must be logged into the site to create or edit records, however.
Times and Days
Please remember that when you're look at the Calendar, times will be shown in your own local time (assuming that your Time Zone and Daylight Savings are registered correctly in your Site Profile). When you are creating or editing a record, however, times must be expressed in terms of Universal Co-ordinated Time (UTC), which is not subject to Daylights Savings, and all Time Zones are expressed as an offset from UTC.
For example, Eastern Daylight Time is UTC -4. British Summer Time is UTC +1.
Days not Rolling over or Rolling Back
One disadvantage to this method of displaying records, is that times that cross Midnight will not cause the day to "roll over":
For example if you're in New York and want to set up a time slot of 10pm (EDT) on Thursdays, that's 2am on Friday in UTC terms, because you're 4 hours behind UTC, but there is no easy or elegant method by which I can cause the day to "roll over" to Friday for people who are in Europe, Africa or Asia if you specify Thursday, or for the day to "roll back" to Thursday for people in the USA, South America or Canada if you specify Friday. Some people, therefore, are going to see the "wrong" day if the time is on a different day from you in UTC terms, whatever you do.
This is because we're not dealing with actual dates here, so the normal method of doing date arithmetic that is employed throughout the IAC site will not work here. Sorry about that, but there isn't a huge amount I can do about that without writing a load of code which will slow the page down considerably (and which I don't currently have the time to do).