You may be confusing email with instant messaging. As a channel for exchanging documents (as opposed to messages) email (and desktop email especially) is still indispensable.
Don't think so. Desktop mail is not much different from web mail when it comes to exchanging documents. Heck, these days with all these web apps around, you don't even need to use a desktop app.
I disagree. Browsers still do not let you embed your data (files) into submitted pages conveniently. We are still unable to easily grab a file and email it to 5 people using webmail. Moreover, even slightly complex documents (i.e. with MORE than just text with basic formatting) are pain to email/consume with webmail.
Or try emailing me a snipped of source code preserving syntax highlighting, formatting and fonts&colors.
Webmail is in stone age when it comes to feature set and usability. It has one HUGE advantage (mobility) but that's about it.
Most people are fine using it because most people, understandably, are not sending each other "documents", but I would describe a typical personal email rather as "short message".
You are 100% right about lack of security though.