I used my Carnet for Syria, Jordan and Lebanon, but I met a German rider who made it to Syria, at least, without one. Well, he had one, but he refused to use it unless they demanded it from him.
As for the visas, it depends a bit on your nationality. I'm Canadian. I had no problems getting visas at the border for all these countries. Now, you may meet people that will say you can NOT get your visa at the border for Syria, but this is not true! It can be done. You do, however need to use a major crossing for this to work. And, of course, when they ask you in Syria if you plan to visit Israel (which they will) the answer is emphatically NO.
You're going to have a blast on this trip! Don't miss Crac des Chevaliers in Syria.
When I was in Lebanon they did not allow motorcycles to ride south past Sidon, but it's worth a look. You may need to park it in Beirut and take public transportation from there... with military permission.
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