Replicant

So now to the issue; Is it true that the drivers in Linux 3.6+, that seemingly would operate BCM4334, are libre?
And how much work would it be to pull in these drivers? Could simply updating the kernel from 3.0 to 3.6+ be a solution?

Reading from the link you provided and how I understand it, it should be possible to run it libre "Completely open source host drivers, no binary object files." (you need the binary blobs to make it work at the moment).
Wolfgang Wiedmeyer wrote about getting a newer kernel to work for android on his blog, it seems to be rather hard if not impossible under the given circumstances, https://blog.fossencdi.org/newer-kernel-galaxys3.html.

Garman Hed wrote:

Reading from the link you provided and how I understand it, it should be possible to run it libre "Completely open source host drivers, no binary object files." (you need the binary blobs to make it work at the moment).

That's great, that was my interpretation as well.

Wolfgang Wiedmeyer wrote about getting a newer kernel to work for android on his blog, it seems to be rather hard if not impossible under the given circumstances, https://blog.fossencdi.org/newer-kernel-galaxys3.html.

That sucks. I checked the released source for the Note 2 and Samsung did the same thing there.

Wolfgang Wiedmeyer wrote:

The goal should be to have an easily maintainable kernel that makes future kernel updates possible. In the next post, I will explain why going mainline is the only solution and how far I've got with getting the mainline kernel working.

This sounds optimistic and I'll be impatiently awaiting the next blogpost.

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If we cannot upgrade the kernel, is there a possibility we could backport the specific driver added in 3.6 to 3.0?

Also, seems I forgot to tag this with device as "Galaxy Note 2", and the Wi-Fi tag.

The Galaxy Note 2 has a Wi-Fi chipset called Broadcom BCM4334, but Replicant does not include drivers to operate this chipset.
Thus, there is no Wi-Fi with Replicant on the Galaxy Note 2.

On all Replicant supported (and partially supported/work in progress) devices the integrated WiFi doesn't work. This is not because of drivers issues, but because theses free software drivers do require a non-free firmware to work, which Replicant doesn't redistribute because they are non-free.
Since Replicant keeps the free software WiFi drivers, the WiFi fails when it tries to load the WiFi firmware and doesn't find them.

WiFi chips and cards that do not require non-free firmware do exists, but they are used not in smartphones (with the exception of the Neo Freerunner, which hasn't enough RAM to run Replicant).

I also don't know for sure if any tablets that has WiFi chips that don't require non-free firmware do exists.
Some have an ESP8266 chip, that might be worth investigating.

I tried searching for drivers for the BCM4334 and found this; https://wireless.wiki.kernel.org/en/users/Drivers/brcm80211
I'm unsure what this page is actually saying so correct me if I'm wrong,
but that link seems to state there are libre drivers for the BCM4334 as of Linux kernel 3.6+; "Supported in 3.6+" 

Running uname -a on my own Note 2 with Replicant 6.0 0002 shows kernel version 3.0.101-replicant, which would in theory be too old.

Not necessarly, see compat-wireless here: https://wireless.wiki.kernel.org/en/users/download/hacking