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Simple:

Does a Gorkanaut gain the benefit from furious charge if he is already S10 due to the Claw of Mork(or is it Gork)? Can he get to S11?

If it has the special rule it gains the benefit.

Whether or not that benefit is useful or not is a different story.

The strength for the Claw is set, and not based off of the model in any way, it would seem.

You cannot go above S10 without a specific allowance to do so.

If the klaw is destroyed, which is possible late in the game if the Gorka/morka-naut is still alive the model is only base str 8 so it would give it str 9 on the charge still without the str 10 klaw.

Keep in mind that sometimes redundant Rules are applied to a Model for sake of 'fluffy reasons.'
Most common occurrence of this is if the Model in question belongs to a group which usually have that Rule, such as Eldar and their Battle Focus.

^ what he said. Eldar Jetbikes have battle focus, but can't run. It's just there for consistency.