Sometimes.
Usually it’s just the “homepage” or the “Startup page”, not the “New Tab page”.

I think Vivaldi is the only one right now, that lets you set an arbitrary New Tab page.

I also change the default search immediately with any new browser.
I switch browsers a lot, and there are close to a dozen defaults I change right away.

Don’t know.
I’ve had a little extension to make home and new tab be an arbitrary URI of my choosing.

Bad title. Not a delivery.
Just a drive around the campus to see if it works.