Picture/Video messaging (MMS) does not charge you megabytes to send/receive messages, however it is a service that accesses Verizon's data network. You are currently unable to send/receive MMS because there is what's called a National Access Block on your account. This is a total block of all your data services on the phone, which is why it blocks your MMS.
Megabyte usage isn't exclusive to the "web" and the term "web" can be very subjective. There are several ways your phones can access data that will charge you megabytes. Pulling up the Mobile Web application is what's truly considered browsing the "web". The Mobile Web can be accessed by a shortcut key on the keypad or thru the Media Center menu under News & Info. Also in the Media Center are several options for Pictures, Music, Games and Browse & Download. Each one of these menus will have a Get New selection (ex; get new applications or get new pictures) and when you click on it the Media Center catalog will load. Browsing the Media Center catalog will charge you megabytes even if you don't purchase anything. Browsing the Media Center catalog can also be referred to as "browsing the web" but in actuality it's not the internet as you and I know it.
Also, some types of applications you might have will charge you megabytes for their usage. So you'll need to figure out which apps you have to see if an application such as a game or VCast Song ID is the culprit. Anytime you access the app and you see a little phone icon with arrows going back and forth then you're accessing the data network and getting charged for megabyte usage.
There are several blocks that you can have on the account without it affecting your MMS: App Downloads, Ringback Tones, VCast Music, VCast Videos, Web Access, VCast Mobile TV and Premium SMS. With all these blocks in place the phone will still have access to the Media Center catalog (the Get New link) which as we know will charge you megabytes just for pulling it up and browsing.
It may seem like a cat & mouse game and on Verizon's part it may seem a bit sneaky; but have a good discussion with the family, let them know what they can and cannot do and it's something that you can control. The megabytes will only show up if something is accessed, that's why it's a pay-as-you-go feature. If it isn't used then it isn't charged.