A. Yes, please complete the digital paperwork as presented.
A. There is no fee for using the Source Engine for your free mod, but there may be licensing fees for included tools if you charge money for your game.
A. Yes, but there are a few requirements:
A. You'll need to copy the exe and content from the Valve Source Engine game you have built your mod off and wrap that up as your application.
While you should include all the necessary Source Engine dlls and content from the base Source game that you need for your mod to run, all Source Engine mods must be configured to require ownership of a Source Engine base game in order to play. This needs to be configured by Valve.
A. Use the CAchievementMgr object after configuration achievements via your Game page on the partner side. For more details, please see the Achievements Documentation (must be signed in to the Steamworks Developer site, which may require accepting the SDK access agreement if you haven't already).
A. You will need a new dedicated server appid so that you can create depots that are for a dedicated server. Once you've got an appID for your base game, please post in the Steamworks Development Discussions and request an appID for your dedicated server.
A. Yes. However, please make it clear on your store pages that your mod is a mod and not created by Valve.
A. The best place to start is the Valve developer wiki here.