I can understand the charge for changes but this isn't a change. They should have either set up the folder permissions correctly from the get-go or ensured that you had the ability to change them as needed. My guess is that the server is incorrectly configured in such a way that you (ie the site) doesn't own its own folders and, therefore, does not have the right to change their permissions. So it's a clear oversight on their part that they should now rectify.
I think it's pretty bad to force me to have to pay for items that I feel are not too difficult to do myself.
I agree. It's important that site owners take responsibility for their own sites and learn how to do some of the general site ownership tasks - even if that does sometimes mean a stiff learning curve. It's just possible that they've tied things down tightly to try and stop site owners from making major mistakes - thus bringing whole sites down - but that's a very fine line to set up and removal of too much control is not a good thing long term.
See what they say about this issue. Point them at this thread if you think it will help. You never know - they may fix it quickly without any fuss.
Do you know of a way for me to change their analytics code to mine?
Looks like it's just been added to your theme's footer.php (which is the standard way to add it). I've placed a copy of the existing code at http://pastebin.com/wsmVimRe. Just edit footer.php in your current theme and replace that code with your own from Google.