Is is possible to pass a custom argument to android market so my app receives it on first launch?

You can publish a link like that
http://market.android.com/details?id=your.package.name&referrer=your_referrer_parameter
After user clicks this link and installs the application your broadcast receiver will receive a broadcast com.android.vending.INSTALL_REFERRER with "your_referrer_parameter" value.

More info:
http://code.google.com/mobile/analytics/docs/android/#android-market-tracking
Get referrer after installing app from Android Market
Get Android Google Analytics referrer tag

AFAIK what you're asking is not possible. The market only delivers the APK files to devices. However depending on what exactly you want to do there are probably many different work arounds.

Here are a couple thoughts:

If you want the market to deliver a custom argument that is unique for every user, then why not have your app connect to a server on the initial run and download that argument? Even if the market could provide the argument it would have to get it from you and you would presumably have to setup a server to provide the market with the argument.

If you simply want the app to know wether or not it is running for the first time you can do that using a SharedPreference. Query if a preference like hasAppRunBefore exists and if it doesn't then you know the app is running for the first time since install. Then set the hasAppRunBefore variable to some value indicating that it has run before. This implementation will allow users to uninstall the app and reinstall it and after each reinstall the app will run for the first time again.

Another option is a combination of the first two. You can have the app connect to your server and provide the server with the device's UUID then the server can check if its seen that UUID before. If it hasn't it provides the argument otherwise it doesn't.

If you truly need each APK to be different for each device you can setup a server that when a download request is received it compiles a new APK and provides a link to download that APK. That will allow you to generate a new and unique APK for each download. This will however require you to distribute the APK yourself as the Android Market doesn't currently provide this functionality.

I would go ahead and have the website that redirects to the market also push a file to the client. the file can be named something like "yourapp.info" and contain the data you need. Once your app starts, it can search the SD card (it should reside in a couple of well known directories, aka /sdcard/Downloads ) and read that file. There are no access restrictions on the sdcard.

Regarding a Cookie in the browser: I'm not sure that you could access the cookie from just any other app - (check this: blog.watchfire.com/files/advisory-android-browser.pdf - it's not possible to access the cookies) so I think that route will be closed.