I am not sure about the email router issue, but I have some ideas on the outlook problem. You may need to revisit the configuration of the email router along with firewall settings and such.
If this is CRM online I would uninstall and reinstall windows live essentials as they tend to clash and this fixes the problem.
If not, verify that your time, date, timezone settings are correct on the client and the CRM server as it tends to interferee with client - server WCF communications with the CRM services if the client and server are too far off.
Lastly, just to make sure your email router and outlook clients are being set up properly I would refer you to the implementation guide for exact tutorials.
Implementation guide link:
http://www.microsoft.com/downloads/en/details.aspx?FamilyID=9886AB96-3571-420F-83AD-246899482FB4
I hope this helps!
Jamie Miley
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