Does Qatar have a LNG terminal?
Yes, Qatargas has multiple LNG 'trains' and has been operating for decades. It is the world's largest provider of LNG (Australia is #2).

Why is Syria necessary for LNG?
It's not. But global demand for nat-gas continues to increase and Qatar will clearly benefit from supplying that demand. A pipeline between Qatar and Turkey would allow Qatari gas to flow to Europe via the Nabucco pipeline and/or LNG export terminals.
This would compete directly with Russian gas and surely explains why Syria has no interest in facilitating any such pipeline. 
Meanwhile, options for a pipeline to the Mediterranean are extremely limited because it must cross Saudi Arabia and then Iraq or Syria, etc.

Furthermore, USA fancies itself as an exporter of LNG and stands to benefit from a squeeze on Qatari gas. So the web is indeed a tangled one.