Q1 Fixed IPs are becoming standard on more and more connections but yes some providers still do charge.
While a dynamic IP may appear to be 'sticky' and not change it is possible that if the router is rebooted or has to resynchronise with the exchange the IP will change, selecting a provider who either charges for a fixed IP or includes it for free guards against this possibility.
Q2 to do what you wish to do you will either have to:
1. If you are going to use Network Address Translation which is the default setting for most routers (devices on the LAN side are typically allocated a 192.168.x.x address) you will need to place the camera or it's control device in the Demilitarized Zone (DMZ) so that you can access it remotely by putting the router's IP address into a browser.
2. A better solution might be to request an IP block from your ISP (which they may charge for) and give it it's own IP address which would be useful if you had more then one camera control board or addressable camera.
if you really just wanted to use the dynamic IP some routers support the Dynamic DNS service run by the American company Dyn http://dyn.com/,
allows you to assign a hostname to the connection and the router updates the companies servers when your IP changes.
Virgin (ADSL) => Namesco => Newnet => O2 => Plusnet => Zen => Newnet => Zen => Freeola => Vivaciti (using O2 Wholesale DSL) => Xilo (C&W Wholesale)
Note: I don't lay turf for anyone. astro or otherwise, all views and opinions expressed are my own based on experience.