From the top of my head, and really rudimentary. I'll be more elaborate once I get home.

Yes, you need to package the map (NOT the mod) by clicking the Mod drop-down menu and choosing Package MOD. In the window that appears, check the "Package this level: [Level name]" box and hit OK. Then copy the entire level folder to your FS/Levels folder.

Once you've done this, go to your newly copied Level folder in your FS/Levels folder and copy GamePlayObjects from the Editor sub-folder. Put it in a folder called 16 in a folder system that says something like gamemodes/gpm_cq/16 (you'll have to create these folders yourself) and copy gamemodes into the server.zip file.

Next, open the map's Init file. You'll notice that all the classes are blank. Copy these from another FS map's init con file, then put the newly edited file (with class names) into your server.zip file. You'll also want to copy this file to the level folder you're editing, and make it read-only, otherwise the editor will swipe the classes every time you save the level.

Finally, get the .desc file, as Zyth says. Copy one from some other FS level, open it, and change every instance of the original map name to your map name. For example "Endor_Clearing" to "Your_Map_Name".

Try doing these steps, and then get back to me. Again, I'll be more precise when I get home, 'cause I know that getting FS/BF2142 to launch levels from BF2 Editor is a ***** the first few times. You'll get into the routine pretty quick, though, no worries.

FYI, you shouldn't have to copy GamePlayObjects or even edit the .desc file if you setup your LayerInit.con file.  This saves alot of tedious copying and editing.

Just grab the LayerInit.con file from bf2editor/mods/bf2 and place it in bf2editor/mods/FirstStrike/.  I edited mine and took out references to SupplyLine, Coop, and SP.  This creates less clutter in the Gamemodes folder.  As I'm only interested in creating 16-64 player Conquest maps my LayerInit file looks like this:

LayerManager.CreateDefaultFilter gpm_cq Conquest

LayerManager.CreateDefaultPlayerFilter 16
LayerManager.CreateDefaultPlayerFilter 32
LayerManager.CreateDefaultPlayerFilter 64

Setting this file up will also write the appropriate gamemodes to your .desc file.  The lack of these gamemodes in the .desc file is specifically what causes a map not to show up in the list.  You will no longer have to copy your GamePlayObjects either.

Your CPs are probably not showing up because your GamePlayObjects.con file hasn't been copied from your Editor folder to your GameModes folder of your map.  If you don't have your LayerInit.con file setup then you'll have to manually copy your GamePlayObjects.con file from your Editor folder to GameModes/gpm_cq/16/.  However, setting up your LayerInit.con file saves you from all this.  Read my above post to see how to do that.   

Also, as Safe-Keeper mentioned, make sure to copy the classes from the Init.con file of another FS map to the Init.con file of your map.  Specifically the area between the rem ---- LevelSettings ----- part.