Greetings opportunist,
Welcome to the forums.
I agree with mostly harmless, I've never installed an app that required me to factory reset to get rid of it.
I use Skype with no issues.
I don't synch my music, so I can't reply to that one.
Apps can be automatically reloaded to your phone if they are synched to your Google account. There are apps that do this function as well, titanium and my backup pro are a couple.
I have a feeling that this might be device related vs app related issues. If you have a mid to low end device, these tend to cause you more issues with apps and such, just because they don't usually have enough power.
As stated above, telling us the name of the device your using is the first step in trying to solve your issues.