This may sound crazy, but Texas would go into the British orbit because during that time in one way or another; because most of the world was within the British Orbit. British business and industrialization was constantly searching for new raw materials markets, and controlled the majority of seabourne trade.
The big question would be how Mexico and the US relates to Texas. If the Texan government truly wants to keep their independence, they have to avoid war with both (unless Mexico really does totally collapse). Britain is the only one that could stand up to the US.
Very unlikely. Britain and the US already had a very profitable cotton trade going on, the UK has nothing to gain by risking that to prop up a dead-end secessionist country with nothing to offer. The only reason Texas contacted the UK in OTL was to scare the US into annexing them (Texas), not out of any wish to ally or get close to Britain.