|it is a Cat 5e cable (although the sleeve is white)|
The color of the casing doesn't matter. I use both blue and white at work. Typically, the blue is for data, the white for telephones.
All that matters is that it says it's Cat5e on the casing.
So I guess all I have to do is properly wire an RJ-45 connector to the wires and make the proper connections in the wiring closet.
Correct. Double check the A and B wiring standards on the internet (google "network cable wiring standards") and use one or the other. I generally use A but it doesn't matter which you use as long as you punch both ends with the same standard.
In the wiring closet, are the cables just hanging loose with RJ45 (male) ends on them? Or, are they punched down onto a "Network Patch Panel" (google that for an image to look at). If they terminate in a patch panel, you'll need a special "punchdown" tool to make that connection.
Then I can connect a lan cable from a spare port on the router to the new RJ-45 jack, right?
If it will reach, why not put and RJ45 end on it and plug it directly into a LAN port on the router?
RJ45 ends are male. The ports on the router are female. So lets say you put an RJ45 end on the cable that was being used for the phone. If it's not long enough to reach the router as I recommended above, and you try to connect it with a patch cable you bought, you end up with two (male) RJ45 ends looking at each other.
The resolution for that is to buy a "coupler".
The coupler is a simple little, inexpensive, device that has two RJ45 (female) connectors. It's designed to join two cables with male ends and usually only costs a couple dollars.
Alternative to the coupler you could buy a "keystone" style RJ45 connector but that is in and of itself more expensive than a coupler and requires a special "punchdown" tool (as compared to the crimper one uses to put on the male RJ45 ends). I'd buy a coupler!
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.