It really does depend on your intentions.
Are you building an "original as you can" trailler queen? Maybe a "looks pretty correct weekend show car? Or maybe a useful fun weekend toy?
In response to your questions:
- Are replacement carpets equally as thick?
IME original carpet is quite, actually (very) thin..... There are still a few bits of what I think to be original carpet in my 63 and I would describe it a kind of thin felt.
Replacement floor coverings come in a million (almost

variations. Did your particular model have carpet (most early ones had rubber mats)? You would be incredibly lucky to find "original" replacement carpets. So, again, does the replacement have to be right, look right or usefully replicate that which "might" have been there?
- Was underfelt originally fitted at the factory?
Yes.
Second, regarding sound deadening..................
- Was the car originally fitted with any form of sound deadening?
Yes, from the early 60s (probably from the beginning) there were fibrous sound deadening pads under pretty much all of the trim (floor, on the roof under the roof lining, behind the door and dash trim... pretty much any vinyl trim had a sound deadening fibrous lining behind it.
- How noisy is the car in use?
If you're inexperienced with Minis? the answer will be "very". Minis were a very advanced design in their time... however, NVH was very much an emerging technology...On long (couple of hours) fast (70-80mph) trips I use noise cancelling headphones. Around town, I listen to the radio.... Its not a Honda but back in the day, we drove them to work, took girlfriends to the movies and the beach ... its a car
BTW looks like you've got a very nice and well preserved survivor there... should be a very nice example with not a great deal of hard work.
Cheers, Ian
PS.. Just a personal opinion. I stay well clear of gluing that bituminous modern sound deadening every where. You aren't going to turn your Mini into a concert hall and it gets in the way when you need to access the floor, firewall etc during maintenance.