Yes, Gig over copper is 250M per pair, all pair are used to go both directions at the same time (full duplex); i.e, all four pair are transmitting -and receiving- at the same time.
Either Gig or 100Mbps will go equally slow over poorly done or improperly terminated cable. The higher the speed, the more critical it is to do it right. Poorly done Cat6 will suck just as bad as poorly done Cat5e (or Cat5 for that matter).
Gig is fully supported on Cat5e. Many/most Gig is supported on Cat5 (with a strong recommendation for at least 5e). Cat6 will provide a little more headroom, and will support other structured cabling functions (i.e., video) by delivering a better quality signal.
Many vendors will support automatic configuration of straight-through / crossover ... but some don't (some/most/all Cisco, for example).
If you need a crossover, you need to cross all four pair (versus just 1&2, 3&6 with 10/100).
A suitable PC (server/server-class) can easily approach Gig speeds (mid-900 Mbps or more), many on several Gig interfaces concurrently. A suitable PC won't have a 32 bit PCI bus. There are other tweaks necessary for full throughput (like jumbo frames... not supported on most consumer equipment).
Most switch-switch, switch-router, or router-router trunks can do "full" gig, no problem. In a commercial environment, fiber is more reliable and more likely to realize the highest speeds, especially in shops where the networking "professional(s)" wouldn't know proper termination and cable certification if it bit 'em in the ass. They are at least smart enough to know they aren't properly educated or equipped to terminate their own fiber, so they buy the jumpers and have professionals install the plant.
FWIW
Scott