What tool would you recommend to monitor the connectivity status of a machine, this is if a given machine it is able to connect to some web servers over time. It should be able to log the status.
There is a long list of freeware at http://ping-monitors.qarchive.org/
When we had to do something similar, we just mocked up some VBS script to attempt to connec to the machines we needed to log. Obviously behind the firewall, on the same domain. Dumped the logs into Excel. Quick and dirty for some network diagnostics, but not a long term solution.
I tend to use Nagios and OpenNMS to monitor large batches of servers (and in the Unix environment, not windows). However, some pure windows-only shops I've worked with have really liked using What's Up Gold. Alternately, a combination of a quick perl script, the LWP library from CPAN and the scheduled task manager would probably do the trick too.