Is it possible to use an iOS browser for Mac such as Phone Disk to apply a jailbreak tweak to a non-jailbroken iOS device?
No, you cannot, for a number of reasons:
With that being said, you're going to have to Jailbreak in order to install Jailbreak tweaks.
Note: I haven't used Phone Disk, but iFunBox is pretty much the same thing.
There are some hacks, such as a hidden panorama mode, which reportedly work without a jailbreak. The App to use on your Mac or PC to make the necessary changes to tweak your iPhone is called iBackupBot. I'll bet if you dig around those files, you may find more cool things.
Remember, a jailbreak consists of a firmware image which has been modified, presumably at the binary level. (Odds are that jail breakers don't have access to iOS's actual source code. So, they'd use tools like binary/hex editors.) Hacks that require binary changes can't be used without changing the code that the phone executes. Changing the binary code is the essence of a jailbreak.
However, sometimes, as in the case of the hidden panorama mode mentioned above, apps (or even parts of iOS, more below) will read "settings" or "preferences" from external files. In those cases, simply supplying another value by editing property-list files may cause the app to act differently.
Apps that tap into the Settings app to store their settings write those settings to the phone's persistent storage as property-lists, which are simply a special form of XML.
In the panorama mode example, the Camera app looks inside the relevant property-list file to check for a "flag" or a switch which will turn panorama mode on or off. There just isn't any user interface for turning the mode on or off. (For clarity's sake: The XML to turn panorama on is not in the same place that most settings are.)
Using the tool mentioned in the article, I've noticed that technically, which icons are shown on SpringBoard is also editable. There are tens of property-list files which configure the system, you just need to have access to them and know what values might be useful.
iOS uses a lot of "Property Lists" and XML, and knowing what does what can yield interesting results. Just bear in mind that most of the "useful" hacks are either available via Settings, or only via a binary jailbreak.
No. Tweaks and items found outside of the App Store are not given permission to run on iOS. Any program must pass a series of checks before it is allowed to execute. Jailbreaking removes (most of) these checks.
Moreover, they all require various frameworks and libraries also not included in Apple's mobile operating system.
Lastly, you may only copy things to your user account. Tweaks and apps found in Cydia require root access (and that can only come by way of jailbreaking).
Some of these old posts contain misinformation. You can install cydia apps/tweaks and maintain a sandbox.
The weblin and ipwnstore are cool stuff. Most of my tweaks still do rely on being out of the sandbox.