Hi Tim,
Greeting from Peggy, Top China Travel.
I'm glad to help you on your Lhasa tour and now will list my answers one by one in the following:
How long does it take to get a tibet permit?
A Tibet permit application may need at least 20 day to deal with.
Can I book a hotel called Tibet Gorkha Hotel?
Because of the safety consideration, foreign travelers could not travel to Tibet by themselves, only China local travel agency can arrange the tour for foreign travelers.
That means you can choose Tibet Gorkha hotel to book, but you may need ask China local agency to book for you.
Does the trip cost include air transportation?
No, the 4 days Lhasa tour on our site excludes the air fare.
About Tibet Travel Permit application
One more thing should let you know that your passport and visa scan copies are needed to apply Tibet permit. And better to apply the permit at least 20 days before your Tibet tour. The ealier the better.
As your Lhasa tour is between June 14 - 17, a month later. If you decide to book the tour with us, we can help you to deal with Tibet permit application immdiately so that you can continue your Tibet tour on time. I am pleasure help you in any moment!
Feel free to contact with us in any moment by mail if you have any question on China tour service.
Peggy
Have a nice day!
Best Regards
Peggy