I am trying to determine how others are configuring their proxy authentication info. In the MDS, there is a setting, "ALLOW THE MOBILE DATA SERVICE TO SUPPORT HTTP AUTHENTICATION." The range seems to be from 0 to 86400000 ms (24 hours).

1) What are other admins setting this to?
2) What exactly does this control? Is this the time for how often a user would be prompted to go though the proxy on their BlackBerry Browser?

It's the length of time that authentication information is stored on the proxy before its removed.
In a security concious environment, maybe 15-30mins might be long enough to have a connection this open. Got to remember that browsing on a device is not the same as on a desktop, so some consideration must be given to how long it takes to initiate an HTTP REQUEST and receive a response.

Right now our users are prompted to enter their authentication information to view external sites on their BlackBerry Browser every 2 hours or so. They don't like this and want the time extended. Is this the setting we change to allow for up to 24 hours browsing before being prompted to authenticate? Does this cause any strain on the MDS?

Browsing activity is session based....therefore by default would use BES/MDS resources.
Extending the connection timeout does not.
Does this behaviour occur for any/all sites or previously visited sites (which would be a cookie issue)..?

Any / all sites. Every 2 hours we are prompted to enter our Windows username, domain, and password.

Possible solutions: Under - Mobile Data Services->Edit Properties-Click HTTP.
Have a look at "HTTP Handheld Connection Timeout" and "HTTP Server Connection Timeout" values.
The first one describes how long the connection waits for the HH to "send" data.
The second one describes how long the connection waits for the request from the sending server.
You could also look at the "Support cookie storage" value.