Is there anything special about the "Applications" folder?
From some experimentation here (not official documentation, so interpret accordingly), it appears that


You can run most applications successfully from any folder, but
Yes, there are some special properties of the /Applications folder.


I include the caveat "most" in point #1 because any application could be written to contain a script that (unwisely) had a hard path coded in it to /Applications/App name.app/Contents/whatever, in which case it would break if run from somewhere other than the /Applications folder. Bad design, but certainly someone could write that way. From experimenting, I did not come across any applications that I use that didn't work when from from various folders in my ~ directory, but others report that such applications exist, and include MAMP.

However, I have found that Applications downloaded and run from various points in ~ do not show up in Launchpad. In contrast, as soon as an app is placed in /Applications, it shows up in Launchpad, and remains there correctly pointing to its current location even if you subsequently move it back to a location in your home directory. I have not been able to determine whether Applications need to be in an officially supported /Applications or ~/Applications directory to properly register Services with the operating system.

So my tentative answer here is yes, you can run applications from outside ~/Applications, but there is at least a little that is special about that particular folder.