As for their only carrier, there's no way they would send it 6,000 miles to see if they can actually launch and land aircraft.
They would not be able to bear the embarrassment of towing it home.
Plus, the know that if Obama loses his mind and goes all butch on them, if the order went out for whatever reason, they could not protect that carrier or it's feeble escorts.
Many Americans take the US Navy for granted in that they don't appreciate their capabilities. Right now Russia has 40 or 50 aircraft on an airfield in Syria, static target.
The US Navy can put 75 aircraft anywhere, per carrier. Plus, these new Burke Class, 9-10kt "Destroyers are amazing.
120+ Vertical Launch Tubes that each carry 1-4 missiles of several varieties. And the best Radar/Fire Control/Targeting system ever devised.
When they travel they establish a 300 miles envelope around them, to include "up".
Your perceptions and the reasons you state are the exact reasons why China would opt to become involved. They strive to become a world superpower, yet lack battle tested experience and respect in the world community. They have spent billions to create a modern military. Several more carriers and scores of modern ships are under construction. We constantly underestimate their capabilities and challenge them even in their own backyard. If they conduct a successful “shock and awe” campaign like the Russians have, we will give pause to challenging their moves around the Spratley Islands and other places. They also want to show that they have the political will and military might to protect their interests. Its not about our military capabilities. They know we will not have the will to fully use them. They know we will not send our carriers into an area in close proximity to potential hostiles who are conducting active military operations. Small as it may be, they are doing their first force projection and area denial maneuver. They don’t think we will challenge them for a piece of dirt where we have no legitimate interest. We are not gonna start WWIII with China and Russia over Syria. We all know BO will never be that butch and the public would openly revolt if he did.
And should their carrier break down, they will simply airlift a million Chinamen and spare no expense to fix it.