A wireless router like the WRT54G uses NAT and packet filtering to prevent incoming connections, unless you configure the router to permit them. However, you should be able to initiate outbound connections of any type through the router, including SSL sessions to your bank's Web site and Hotmail.
To make sure that outgoing SSL connections are permitted through your Linksys router, use the Advanced tab on your router's GUI to check the Filters (or Access Restrictions) page. Port 443 should NOT be filtered, and "Disable Internet access for listed PCs" should be checked WITHOUT your notebook in the PC list. If "Enable Internet access for listed PCs" IS checked, then your notebook MUST be in that list. Use your router's Outgoing Connections Log to see whether connections are being placed. If SSL connection requests are making it through your router, then perhaps something between your router and the Internet is blocking SSL (e.g., a firewall or your ISP).
To accept incoming connections used by some Windows Messenger features, use the Advanced tab on your router's GUI to configure Port Forwarding (or Applications and Gaming). Enter the TCP or UDP port number you want to receive connections on, and the IP address of the PC behind the router you want to handle those connections. You must also configure a static IP address on the destination PC and make sure that IP is not within the range your router's DHCP server uses. For example, suppose your router's DHCP server assigns from the range 192.168.1.100-150, and uses 192.168.1.1 as its own LAN address. Configure your notebook's wireless card with the static IP 192.168.1.2, and then create port forwards to receive incoming Windows Messenger requests. See this Microsoft article for guidance on ports used by various Messenger programs. Also, make sure the notebook is not running a desktop firewall that would prevent it from accepting incoming connections, such as ZoneAlarm, McAfee, or the Windows Internet Connection Firewall.