Heres where things get a little tricky.
Dropbox isn't an app tailored to your needs. It wasn't designed to edit PDF's or files, although, it can view them and that functionality can one day exist, but it's intent as a business is to offer customers a place to store their documents that can be accesabile across all their devices. With Apple, and developing for them, you are constrained to your app and it's bundle of resources. So for example, Dropbox. Dropbox is a application stored on the device, that can only write and save files to the applications bundle (think folder for bundle) and no where else on the device, this is for numerous safety precautions implemented by Apple. Makes sense. The drawback, is that the Developers have to make the features available that you want, not Apple. If the Developers want you to be able to edit the PDF then they have to explicitly code for that feature, otherwise they can share with extensions a file/document to another app that already does that. Again, since Dropbox is in the business of storing files that would be their motive to not allow. Things can change, you can always submit a request, but it's not that easy developing things, it takes a lot of work and man-hours.
So to answer your questions :
1 ) Yes - a copy is sent to the application you elect because, as stated before, an app can only directly effect the resources in it's bundle (app folder) on the device and no where else so Dropbox has no choice, it's an Apple regulated process.
2) The new file is stored in the other applications bundle
3) iCloud works similarly but not really. Since iCloud is developed by Apple they obviously have more freedom to do what they want. When you store a document in the cloud it stays there and you can manipulate it in the cloud, or download it to the device and manipulate there as well and re-upload it. Other apps can do this too, but they must code for it themselves, it's not just handed to them. Apps can take months to build and are a lot of work because of all these work-arounds, so don't take them for granted. I'm surprised stuff sells for just .99 cents when they have been in development for 12 months.
Your only alternative is to find an app that allows editing PDF's, and has a share extension for Dropbox, for now, submit a feature request to Dropbox in the mean time to let them know that it's a feature users want and will use and just wait.
EDIT I normally don't like to link to apps or other software but since it's within the realm of the question defined I will post a link to software similar to the goal you are trying to achieve, but as objective and unbiased as I can; I share no relationship with either app :
Edits PDFs from Dropbox, so does numerous other third party apps, you will just have to find whats best suited for you.