To cleanly uninstall Minecraft you need to remove the entire .minecraft folder. If you want to keep your worlds, move the saves folder somewhere else before deleting the folder. You can always drop it back in once you reinstall Minecraft and run it once to re-create the .minecraft folder.

(The program you click isn't really Minecraft itself – it's just the launcher. You can get rid of it too if you want, but deleting it or not is separate from uninstalling Minecraft itself. It doesn't keep any settings or matter much, since it's just a convenience for launching the actual Minecraft program, which is stored in .minecraft. In fact, if you know how, you can start Minecraft without it entirely.)

In particular, I think your texture pack options are causing the crash. I've had that happen before, with an old texture pack crashing the game after I updated it.

I suspect that because, from your other question, it sounds like you were trying to use a 128x texture pack and it was crashing it. By choosing it, that choice is saved in your options.txt file in your .minecraft folder – deleting bin and resources will leave that intact! This option is probably what's crashing Minecraft. (Texture packs that are incompatible with the current version of Minecraft, or which are too large for your computer to handle well, can sometimes crash Minecraft.)

To fix it, open options.txt (using a text editor) and find the line that begins with skin:. It should say "spahx" or something like that. Change the line to:

skin:Default

And then run Minecraft. It should work now, no un/re-install needed.

The culprit

Cannot run program "C:\Program Files (x86)\Java\jre7\bin\javaw.exe"
CreateProcess error=2, The system cannot find the file specified

This is Windows saying that it cannot execute a file at C:\Program Files (x86)\Java\jre7\bin\javaw.exe because it cannot find such a file. The most likely cause is that the file does not exist.

Verify that your Minecraft profile points to the Java installation you want to use. Since you appear to be running a 64-bit OS, you may want to run a 64-bit version of Java for overall better performance.

But I installed Java, more than once even!

all java has been downloaded

Out of security concerns, Oracle's Java installers may opt to remove versions of Java that are outdated or end-of-life. If you've installed version 8 after you installed version 7, the former may have removed the latter (though it should ask you whether you want to remove older versions).

The easiest way to keep an older Java (JRE) installed, if you need to, is to install the highest version first. Older versions will not remove newer versions.