You have to get Thai Airways to do the booking, but I would highly recommend you check out the competence of the local office. In Kuwait for example, TG sucks and made mistakes on my awards booking before. I had to get LH Kuwait to sort it out, PLUS make a trip to Silom TG to verify.
Also... you asked if LH can upgrade you using TG miles at the airport.
I have no experience, but on this point I would certainly phone LH customer service and ask this specific question.
This long link will take you to the page where you can select the country to speak to the Miles & More customer service (but if you have Skype I would call the German numbers to get 100% accurate facts):