Your download speed will vary according to the package that you have from your ISP, so if your package is for a 10Mbps link, then that is what you would expect to get.
No, having a 100Mbps LAN card is not worthless and you can transfer to other systems on your network at that rate, although I must add that you might have an older system if it only has 10/100Mbps for LAN. Almost all come with 10/100/1000 these days.