There are a few incorrect statements here. The link speed is only determined by the device it's connecting to. If your NIC is 1G and your switch or router that it's physically connected to will do 1G, you will be 1G. If there's a downstream link somewhere else that's 100Mb or even 10Mb, you'll be limited to your throughput, but you're still connected at a gig.
The caviat to that is some NICs don't handle 1G very well. Don't hard code 1G NICs. Set everything to auto and you should be good. Your issues could be a NIC issue as opposed to a cable issue. I would be willing to bet that's the case.
Yes, cat5 will handle 1g easily, but as you get longer distances, you may see issues. Within a house/apartment/flat/whatever, any cat5 cable or better should suffice.