I'd recommend having three separate projects in Eclipse: "[game name]-common", "[game name]-client", and "[game name]-server".
Eclipse projects can be hierarchical. You can have add a project the the build path of another (right click on the project -> properties -> java build path ->projects). If a project has another on its build path, it is considered to "share" all of that code. Additionally, you can have settings enabled that will automatically package the other project with an export.
As for the launcher, that's actually a really, really easy problem to solve.
The most important thing to do is to consider the launcher an entirely separate application from your game. Do NOT mix the code of the launcher and the game, it'll make things more complicated and is entirely unnecessary.
First off, you "[game name]-client" project should contain neither the launcher or the applet. Instead, make your game so that it's basically fully-functioning, but have the display-setup code be delegated. So, once it's running, it will take advantage of the already created display, but be fully independent.
Next, make two new projects: "[game name]-launcher" and "[game name]-applet." Both of these will be relatively small. For example, the applet project will probably only contain a single class (your applet class, of course!) that handles initialization like any other lwjgl applet.
Your "launcher" project will be a bit more complicated, but not by much. Build your launcher application in Swing, use a JFrame, blah blah blah you have a nice little window with graphics and buttons that lets you do stuff. Now, simply tie code to an action (i.e., pressing the "play" button) that will close the launcher, and create a LWJGL display. Then, run your game in your "client" code.
Does all that make sense? It has numerous advantages. For example, you'll be able to modify either the game, the applet, or the launcher selectively, and you won't have to worry about screwing up anything else. Additionally, each of your exported jars (the applet or the launcher-based game) will not have unnecessary code for the other launcher system.