You basically got the jist of it all, but thats a little over simplified explaination of it all.
Refugiums serve many purposes, some of which you mentioned. More accurately the point of a sump is filtration, a place to stash all your junk hooked up to your tank so that you will not detract from veiw of the display ( a ugly power head can really kill the ambience of it all as do most skimmers), not only can you hide all your junk in a sump but its much easier to to service/maintain it, and it does in deed give you more water capacity and therfore better water quality stability. Some sumps can have a refugium in them as well. I am not a big fan of these although they do indeed work becuase a sump generally moves water through them at a high flow rate and this is not conduceive to nutrient up take efficiently. Its neither beast or fowl. A dedicated refugium on the other hand will usually have a considerably slower flow rate, this will allow better nutrient up take for macro-algae (for which caulerpa is one type, and not the type I would personally recommend as my first choice for a number of reasons, but none the less is often used and will do the job). It is this enviroment that is a safe havean for copods(sp?), Mysis Shrimp and many others, which will multiply like mad given the chance. It too will give you more water capacity allowing you to have more stable water quality. So there are sumps, sump/refugium combos and refugiums. All a little different in purpose.
At any rate, you will probably need a slightly bigger pump to pump water back to the main display. This is becuase water will have less resistance flowing to a low point such as a sump under the display becuase there is not any head pressure to speak of. On the other hand the pump, pumping water back to a display above will have a good bit of head pressure and therefore have to work much harder to get the same volume back to the display, at least this has been my experience. Might want to think a little more about this before you leap first and look later. Becuase a dedicated refugium generally has lower water flow rates you can use things like Miricale Mud, GARF Grunge Lite, Oolitic sand and the like without it being blow around by water flow. This would be a good thing to add to a refugium if its possible.
For macro-algaes...any of the various types of Caulerpa will work as intended. The downsides are that they have hold fast that will attach to everything in sight making it potentally unmanageable. On top of this it secretes toxins that will impead the growth of any corals you might have which is no big deal in a FO/FOWLR set up. It will also "melt down" on you occasionally sooner or later and in a reef tank with coral would really score high on the "Top Ten Sucks List" to put it mildly. Macro's I would suggest would be "Chaeto", Red Garcalia (sp?) sometimes called Tang Heaven, Red Mangroves( not really a macro-algae but...a nice choice just the same), eel grass if the refugium is kinda deep (15-18 inches of water) and I am sure there are a few more that I didnt specifically mention that would be a sound choice. The Chaeto is nice as Pods will hang out in it big time and reproduce like crazy for you, good for your corals, baby fish if you have any and Manderin Gobies if any. Its easy to grow and very unlikely to melt on you. This will definitely help lower your nitrates too.
In all of my refugiums, and almost all of my tanks have one, I like to add copious amounts of live rock rubble. You can get this really cheap from premium aquatics for about a buck fifty a pound. On a budget? Make some DIY Agrocrete and use empty styrofoam egg crates to make ping pong ball sized rocks and seed them with a good piece of coralline coated live rock. This can be done for about 10 cents a pound! I also like to add some Garf Grunge or Grunge Lite but thats just me personally. A refugium kit with some worms and mini stars and such are also kinda nice to add to. There are a few places that sell these "kits". For substrate I use mainly Oolitic sand. Of course I got a few clean up crew guys in there too just for GP if nothing else. I also put a good number of pepermint shrimp in mine that have reproduced for me upon occasion.
Just some food for thought....
A refugium is a great addition to a tank, and can really give it a boost water quality and food stuff wise. They dont have to be hideious moss buckets, some can actually look quiet striking and have an abundence of life in them.