Which webmail site is this on (e.g., Yahoo, Gmail, Windows Live Hotmail)?

Yahoo

I switched to the latest Yahoo! mail, so I don't know whether we have the same setup. However, it seems to work normally for me. This is what happens:

I click a message in the Inbox, and it opens in the same Firefox tab with an in-page "tab" appearing to the right of the Contacts button. When I start selecting from the top of the message down, holding the left mouse button down and dragging past the bottom of the page, it scrolls all the way to the bottom normally.

Can you describe what's happening for you in a little more detail? Is the problem that when you drag down to the bottom of the part of the page you can see, the page doesn't scroll up automatically the way mine does? Do you have this problem on other long pages, or only in Yahoo mail?

Also, could you test this workaround (it works for me). Select to the end using the 3-step approach:
(1) select the beginning of the text/pictures in the usual way,
(2) use the mouse wheel or scroll bar to scroll down and end of the text you want to select (don't click anywhere in the page while you do this),
(3) hold down the Shift key and click where you want to end the selection, and Firefox should extend your existing the selection to that point.