If you go for a combined modem/router, it should just be a case of inputting your BT Yahoo username and password into the router via it's web interface. There are no compatabilty problems.
You can connect all your PC's wirelessly if you wish, although it is advisable to have one PC wired to the router.
Your daughters PC no longer needs to connect to your PC (unless you want it to) as it connects directly to web through the router.
Your connection speed won't be affected.
You don't have to have the router on permanently if you prefer not to. Just switch it off or disconnect it after each session if you prefer, although if you have set up security adequately, there should be no need to do this.
I have a Netgear DG834G and found it very easy to set up.