Banned Gamertag?

Banned consoles cannot access Live.

Because a console is banned does not mean an individual's gamertag was banned. The two are not linked that way. If a console is banned but the gamertag is not banned, the gamertag can access Live from an unbanned console.

Assuming the gamertag isn't banned, the banned console owner can buy another console, and assuming they abided by the Terms of Use, continue using Live with their gamertag.

Game licences are not transferrable so you wouldn't be able to transfer it to a new gamertage, no.

As to buying a new licence on the banned gamertag, not sure what the point would be if the gamertag is banned from the game? The new licence would still be banned if the gamertag is banned from the game itself.

If your GamerTag was banned, recovering it won't restore services. If your console was banned, recovering the GamerTag on the new console should be OK. If you were banned for using false MS Points, I'd imagine it was a GamerTag ban.

no their accounts are not account banned.

As you say they arealso banned on the forums if their account is banned. For them to post on any forum including "my xbox account was suspended", they need to create a new account, and refer to the banned gamertag in their post.

so in this case, their gamertags are not banned, because they can use their gamertags here.

The reason is simple. MS is on a ban wave of original gamertags that have been stolen or unautherized V1 reset. The gamertag "LOL" was reset using a hack. Therefore MS console banned the user and name banned the gamertag "LOL". As a result of the name ban
the word "LOL" cannot be put as a gamertag, in your motto, bio, name etc. All the other "normal" words that you cannot put in your profile are also stolen gamertags that have been name banned.