Are network cards universal

Not really. Depending on what your definition of "universal" is, there is a good chance you can install one in any given computer and network it, but there are (as with any devices) situations where it may not work in a given situation.

No. A network interface card could be connected through many different buses on a computer, such as PCI, ISA, PCI-E, PCI-X, or USB. If a computer does not have one of these interfaces, a card designed to fit that interface will not work.

No, although this is not really much of an issue, since routers and switches generally support multiple speeds. Just don't expect any improvements in your network if you upgrade your network card but negelect your router.

No. Wireless cards don't use any cabling at all! Still, this is again mostly a non-issue. The most common Ethernet cards and routers these days use a standard twisted-pair cable that is backwards-compatible with slower speeds (10BASE-T, 100BASE-TX, and 1000BASE-T). However, you may need to upgrade your cabling for faster devices, since they are more picky about cable length/ quality than the slower ones.

You'd have to either have megabucks or go out of your way to find them these days, but there are alternative networking cablings, such as fiber-optic (expensive), token ring (obsolete and slow), or coaxial (usually either expensive or ancient).