Can someone please confirm that what I have read is true: that Texas is the only US State permitted to fly its flag at the
same height as the Stars and Stripes. If this is true, what is the reason(s) for it? Thanks very much.

This maybe an urban myth .. have a look at the posts by fosse4 and sequoianoir over at AskFuntrivia.com

Which is the only US State allowed to fly its flag at the same height as the Stars and Stripes?

Here's a link to the "urban legend" exposed:

Urban Legends: Texas

There is one other flag which is always flown above the USA or any other National flag and that's the Olympic Games Flag which is always above all others whilst the games are in progress (The spirit of the games flies above national priorities).

This site says it's the UN flag:

Do not place any other flag or pennant above or to the right of the flag. (The only exceptions are: at the United Nations headquarters, in New York City, the United Nations flag flies above all others; the Navy church pennant flies above the flag during church services.)
Denton County, Texas

see also www.usflag.org

Sorry GT - I forgot to add that UN flag will always fly above the others "politically" but is usually flown (as are all the other countries) at the same height indicating equality of the countries, I was just adding an occasion where another flag took precedence (with all the other countries agreement) and therefore could be flown above any other nations flag. I don't know if there is another occasion where this happens.
Does the Red Cross (or Red Crescent) have priority over a national flag?