Yes, most servers are also connected using ethernet. For most purposes CAT 5e suffices, which can carry up to 1Gb/s (or 125MB/s). Anything above that you need serious hardware to keep up with those speeds.
Fiber optics is more used for long distance (i.e. from you ISP to the area you live in, where it’s split off) or for routers.
Sure, but if your hardware can’t keep up with those speeds having the cable is pointless.
Most ‘high speed’ servers will have trunked 1 or 10Gb lines attached to them. They’re not… strictly all about the line speed though - the idea of a server is to handle volume of traffic, moreso than individual packet ‘speed’. The 10Gb line is all well and good, but unless every hop between the server and client is that same speed, there’s no value in it for an individual consumer; that said, that 10Gb line can also handle a lot of less-than-10Gb speed connections at the same time without slowing traffic.