Yes, any time you transmit data you are charged, and your total usage is rounded up to the nearest MB, so if you use 5KB of data, you will be charged $1.99 for 1MB of data. Simply accidentally clicking on the "Get it Now" or "Mobile Web" functions, no matter how quickly you cancel, will incur this data charge.
As far as I know, you cannot remove these features from your phone, but you can make them more difficult or impossible to access.
Option #1: Go into your phone settings and change your shortcut keys to access something that won't cause you to use data, my favorites are the calculator and alarm clock functions.
Option #2: Log into MyVerizon and block Mobile Web, Ring Tone, Game Ring-back downloads, etc. (This is best used with the above suggestion, because I think you will still have access to browse those catalogs even if you can't download them, and browsing does use data).
Option #3: Call Verizon and ask for a Nationwide Access Block on your account. This will also prevent you from recieving or sending picture or video messages, but it will completely prevent you from using any feature that will cause a data charge.
OR, if you don't like my variety of free options, you can go for the Verizon solution.
Give Verizon $9.99 extra each month for 25MB of data, thus preventing accidental $1.99 charges on your bill.
Since you are using the Rival, you are lucky to have this choice!