Asking for Default App for .pdf Files

Well try this: Select any .pdf file. Press COMMAND-I to open the Get Info window and in the Open With section select your desired default PDF application from the drop down button menu then click on the Change All button. This should assign the default app for all .pdf files.

How many Adobe products are installed that read PDFs? Do you have both Acrobat and Adobe Reader installed or do you have multiple versions of the same application installed?

Is the dialog that appears a Safari or OS X dialog or an Adobe dialog? Adobe apps attempt to install the Adobe PDF Plug-in to replace the PDF reader, Preview, with their own Adobe Reader. It sounds like that may be the issue here. If I am the appearance of the dialog is controlled via one of the preferences in Adobe Reader and/or Adobe Acrobat. You can disable the use of the Adobe plug-in and opt to allow Safari to handle the display of PDFs in the browser.

On the other hand if this is opening any file such as a PDF automatically after downloading, then that's controlled by Safari preferences. In the General section uncheck the option "Open Safe files after downloading ...."