Add new video card to an old Dell Dimension 2400?

The card you specify in your first post should fit into your D2400 PCI slot.  It is sometimes called a "universal" card as it can fit into either a PCI 2.2 or PCI 2.3 slot.  PCI 2.2 and earlier versions allowed 5 volt signalling, and cards specifically designed to work with 5 volts have one key slot toward the rear of the card.  Cards designed for 3.3 volt signalling have the key slot closer to the metal bracket.  Cards that are capable of handling either type of signalling have key slots in both places.

Conclusion is that if you can get the card it should fit into the Dimension 2400 (which happens to have PCI 2.2 version slots and is designed to that standard).  I'm afraid I have only dealt with NewEgg, but I have seen complaints on the forum about Tiger Direct in the past.

With regard to the power supply, I don't think any PCI video card will stress the power supply.  Those cards that need additional power generally have a requirement for an extra plug directly from the power supply.  I've never seen a PCI video card that had one of these.  The 6200 is a bit later than my FX 5200, but it doesn't work all that much faster, and my Dimension 2400 has worked fine with the original power supply for four years now.  I wouldn't worry unless you are also planning to add additional hard drives or other power hungry accessories.

You normally need to leave the bios set to Auto so that the bios will detect the new card and switch to it.  Before you install the card, however, you need to disable the Intel Graphics by going to the device manager and selecting "Do Not Use This Device - Disable" in the Device Usage menu at the bottom of properties.  This allows Windows to load a basic VGA driver that can be used with your new card until you get the specific drivers for the card installed.

Edit:  Oops!  You aren't running Windows.  You will need to figure out the equivalent of what I've written for Windows in your Ubuntu installation.  If you install  the card without going to a plain vanilla VGA driver, your OS may attempt to use the Intel Graphics driver for the card and you will wonder why you can't see anything on the screen.