Is the TMDS intended for long distance transmission?
Short answer is yes, TMDS is intended for long distances, but it depends if you consider roughly 10 meters as long.
Longer answer, the two terms are not exactly comparable.
TMDS is a special kind of 8b/10b line code, a data coding method, which is specified to be sent using CML PHY layer, while LVDS is strictly a PHY layer that can be used to send any data with or without line coding.

Speed. CML (used commonly on TMDS interfaces) allows for faster data rates than LVDS.

Both. For example links between high speed chips can use CML for speed, and also your displays connected with DVI and HDMI use TMDS with CML over the video cables.

Same as 1, speed.