A browser history is a record of requests made while using a web browser, and includes information like websites visited and when.
Tor Browser deletes your browsing history after you close your session.
Domain fronting is a censorship circumvention technique which masks the site you are connecting to.
From the perspective of a censor, it appears like you are connecting to a major service which would be costly for a censor to block, like Microsoft or Google.
However, it does not make you anonymous, or completely hide your destination like Tor Browser does.
For more information see this blogpost about domain fronting.
No need to capitalize.
You can translate this term if it sounds better on your language.
"Tor log" is an automatically-generated list of Tor’s activity that can help diagnose problems.
When something goes wrong with Tor, you may see an option with the error message to "copy Tor log to clipboard".
You should see an option to copy the log to your clipboard, which you can then paste to a document to show whoever is helping you troubleshoot.
If you don't see this option and you have Tor Browser open, you can navigate to the hamburger menu ("≡"), then click on "Settings", and finally on "Connection" in the side bar.
At the bottom of the page, next to the "View the Tor logs" text, click the button "View Logs...".