There's no easy answer, and it may not be possible. But you can try the C**** approach to downgrade it to an earlier version, then try upgrading with iTunes. The problem is the latest jailbreak gets the baseband version out of sync with the iOS version. This is detected as a version mismatch and prevents the update from completing. You may be able to update to a future version.
DFU mode and Recovery mde are the same thing.
What tinyumbrella and similar programs (such as fixrecovery) do is fool the computer into thinking that Apple's validation servers are at a different IP address, either on your computer or on a Cydia server. This server then pretends to validate the installation, fooling the iPhone so it will work without real validation. They do this by inserting lines into the hosts file on your computer. This file is checked before the computer does a DNS lookup to find the IP address associated with a URL. You have to remove any such lines with a text editor. The hosts file is /etc/hosts on a Mac, or %windir%\system32\drivers\etc\hosts on Windows.