Here are the generic instructions (after which I'll address your specific issues):
Install mozilla-apk-cli using NPM:
npm install -g mozilla-apk-cli
Use it to generate a "debuggable" APK for your app from either a source directory or a URL to the mini-manifest:
mozilla-apk-cli /path/to/source/dir/ arbitrary-name.apk
mozilla-apk-cli http://example.com/path/to/mini/manifest.webapp arbitrary-name.apk
(Context-click > Inspect Element on the "Free" button in Marketplace to discover the mini-manifest URL for an app in the Marketplace.)
Install the APK on your Android device:
adb install -r arbitrary-name.apk
Launch the app on the device. Look to the notification area for a notification about which port the remote debugger server is listening on. Forward that port on your desktop, f.e. if the port is 12345:
adb forward tcp:12345 tcp:12345
Go to Web Developer > Connect… in Firefox on your desktop and connect to localhost at the forwarded port. Commence debugging!
Notes:
Now, regarding your specific issues: It sounds like you've made it past app installation but are stuck on launching and running the app. How are you launching the app? For example, are you tapping the app's icon on Android's All Apps screen? And what do you mean when you say that apps "do not start"? Do you mean that nothing happens when you launch the app, or does something happen (f.e. a splash screen appears, perhaps followed by a white screen), but the app itself doesn't appear?
It would also be useful to know the manifest URL of the app you're testing, if it's a public hosted app; or get access to the source code, if it's a public packaged app!
Finally, note this related (identical?) question: How to install packaged app on Firefox for mobile?. I copied the generic part of this answer from it, but presumably they should be consolidated into a single answer (modulo the troubleshooting info, which is specific to the problem you're encountering).