I am using a Sharx 3904 - it was purchased through SebecTec for about $40 less than available on Amazon. I hooked up with SebecTec through a weather hobbyist forum where SebecTec is a regular contributor.
The Sharx allows you to view live video feed or still images through a web browser or cell phone. A static IP is NOT required, the instructions include how to use 'www.no-ip.com' to set up your camera for this. I did not use the service but if I remember correctly the service was free for < 5 cameras. If you needed more than that there was a monthly fee.
The Sharx can also send still images via ftp or to a Dropbox account or all of the above. This can be based upon a set schedule or on a motion trigger or both.
The other IP Camera that gets very high reviews for use as a still "weather cam" is HikVision models. You can get LOTS of information about setting up a camera like you are talking about from
WxForum.net - Weather Web Cams.