Those are dangerous instructions (to set a directory or file to 777) unless you know exactly what you're doing. If an (honest) application wants "777" directories, the author wants to be sure that the program can write to that directory. He was just lazy, or else he'd write a long explanation like this. If PHP is running as "owner" (say, via suPHP), 755 is sufficient. If it's running in your "group", 775 is what's needed. Only if it's running as "other/world" would you have to have 777. In such cases, you normally would try to reset it to 775 or 755 when the program was done writing files (if possible). On a dedicated server, it may not make much difference (there are no other users sharing the server), but it may be better to play it safe and grant only the minimum permissions needed for the application to do its job.

Assuming that there's nothing nefarious going on, and the lazy idiot who wrote the instructions just wanted to be sure his program could write a file in the root directory, 755 is enough. LP's security software (suPHP) won't even permit 777 (world writable) for directories, or 666 for files. Your program will be running as "owner", so it only needs "owner" to be writable.

What's wrong with mixed case names? So long as you get into the habit of being consistent in how you capitalize files and references to them, you'll be fine. Linux servers are "case sensitive", and MyPicture.JPG is considered a different file than mypicture.jpg (even though a Windows server is case-insensitive and will treat them as the same). Or were you referring to some names all lowercase and others ALL UPPERCASE? Even so, it really doesn't matter, unless you expect visitors to be manually typing in the names. Normally no one ever types in a file name, but uses a search engine result or a bookmarked name.

Note that when you change file names (which you shouldn't have done), your search results will suffer for a while, until the new names are indexed and have been up a while. If you don't have a lot of names, a 301 redirect could be done so search engines and visitors don't get 404s, but if there are thousands of names, that becomes infeasible.