How to install bios updates in Zesty (on a Dell laptop)?

Am I doing the wrong thing?

Yes, I think so. There is a page dedicated to BIOS updates and it says about Dell this:

If you are using UEFI and your F12 boot options include "Flash BIOS upgrade", one may download the BIOS upgrade .exe from Dell's website, and put it to your /boot/EFI/ folder. Reboot and select "Flash BIOS upgrade" option. In the dialog select the .exe file you have just downloaded and continue with the process.

For more on Dell specific procedures, please see here.

The feature you're trying is very new, and I've never used it myself, so I can't really fully answer your question. The fact that you managed to update it manually is good, but I thought I'd provide what information I do know....

About a year ago, a new program, called fwupdate, began to be shipped with Ubuntu. This tool is very minimally documented, to the point that I'm not even 100% sure of how it's supposed to work. As near as I can tell, though, you're supposed to be able to download a firmware update file, pass it to fwupdate, and have it applied by a matching EFI program (fwupx64.efi) when the computer reboots. It looks like you've run into a GUI front-end to these tools that's located a firmware update online and is offering you a way to install it -- but that tool is either not working or is working in a way that's leading to confusion. Perhaps you just needed to reboot after you saw the message that disappeared too quickly; or maybe there's a bug somewhere that's causing it to fail for you. Either way, you may want to file a bug report to make the developers aware of the problem you encountered.

As a general rule, I recommend against installing firmware updates unless you have a specific and compelling reason to do so. The reason is that updating the firmware on a device sometimes goes badly wrong. In the case of a motherboard, this can render the computer unusable. Many modern computers provide recovery mechanisms, but at best these require awkward operations with which you're probably not familiar. That said, firmware updates may provide important improvements that can improve system stability or even security, so if you learn of an update, you should review the release notes to learn what it does and decide whether to take the risk of applying the update. This risk is small, but if you run into such a problem, the consequences can be pretty bad, so you shouldn't take the chance if the firmware update is trivial or fixes a problem that doesn't apply to you, such as if it fixes a network-boot problem when you never boot over the network.

I'm running Ubuntu 17.10 (Artful) on a Dell Precision 5510. I was also prompted for a BIOS update and a TPM update, and was also unable to install them through the GUI.

I applied the BIOS update by following the steps in Edit 2 above: I manually downloaded it from Dell, placed it in /boot/efi, rebooted and selected "Flash BIOS upgrade".

I was able to apply the TPM update using the command-line program fmupdmgr. fmupdmgr get-updates will list available updates. fmupdmgr update will schedule available updates to be applied on the next reboot. I ran these two commands in succession and rebooted, and the TPM update was applied.