If you add 4 new relays as designed above, that original older relay is redundant. My opinion leaving it would not harm anything, but an electrician would tell you to yank it out because someday if you do have a problem it could be a cause, where otherwise it would not.
The 14 gauge wire from your new relays to each power window motor should be sufficient. There's a formula printed on most boxes of wire what gauge is appropriate for what amperage and distance. The distance from your new relays to your motors will be less than 4 feet, so it should not be an issue. The only exception is if the wiring itself has been damaged in anyway, then you should replace it. Here are some sites with charts so you can get an idea what's happening:
http://www.crutchfield.com/S-uhH9GJZd4ma/learn/learningcenter/car/cable_gauge_chart.html and
http://www.rallylights.com/hella/SensibleWiring.aspx Basically the longer the distance of the wire then the thicker the wire needs to be to handle the amperage.
For reference, I could not get a definite answer but I believe each power window motor pulls between 12-18 amps.
The new wire feeding the relays should be 10 gauge in my opinion, though looking at the chart I have here as long as the run is under 4' then 12 gauge should work (12 gauge is what the factory used) If you run 12 gauge to the relays though, you could introduce the same problem the factory wiring had when both windows are rolling up/down at the same time, where they then go slower competing for the amperage.
Technically the line coming from the battery was a generalization, you should (though most don't do this) wire the lead to the same place where the factory wires the other wiring to as well, and that is determined by which motor is in your car. I can only speak for a Pontiac motor, in that case the new wire should be added to the back of the alternator using a ring connector. You should also include a fusible link 9" long between your new run to this connection at the alternator, the fusible link is important, again it's what the factory designed. Everything else should be soldered with wire heat shrink, or crimp-able heat shrink butt connectors. I also added insulated spade connectors between the window wiring and relays, that way I can take out the center console with the relays and switches just by unplugging those spade connectors. You can knock yourself out with creativity there. Also, this is just my opinion, you could still wire the window relay directly to the battery post, I see guys who add amps and other stereo equipment do that all the time. Just remember that you need a fuse somewhere in there, otherwise things could get damaged.
