In the cases that I am aware of, the wait time has ranged anywhere
from three months to over a year. An expedited request is possible,
but I don't know what circumstances would qualify for such a request.
Once the travel document has been received, it is quite similar to a
passport. No visa is required for a visit to Thailand of under 30
days. There are a few differences, the most notable being the color
of the cover. Another difference is that it is only valid for 2
years, whereas a passport is valid for 10.
Refugees can become naturalized US citizens 5 years after arrival in
the US. With that comes all the rights and responsibilities of a
native born citizen, including the opportunity to apply for a standard
passport, which is considerably cheaper, easier to obtain, and is
valid for 10 years.
This will also become a larger issue as US-Canada border crossings are
tightening. Only certain documents will be accepted at the border.
Many refugees in the Northern states have crossed the border regularly
in the past to visit their friends and relatives in Canada. In the
future, this will not be possible without proper travel documents.