In regards to the financing of Texas, I think you can get Britain to float an independent Texas' economy as a means of preventing American expansionism. Also Texas is more likely to exist in a world with a set of independent nations in the west like an independent Rio Bravo, California, Mormon state, etc. rather than just as a buffer between the United States and Mexico.
I'd agree here, but I think we can also see Texas remain independent by itself in a scenario like the one I described above. Also, an independent Texas would be the first of these western nations. So Deseret, California, Rio Bravo... and possibly Columbia/Oregon would be following its example.
If we were to keep Texas by itself, I do think its borders will be very much reduced. Even if Mexico accepts an independent Texas, it will try to assert the Nueces - Pecos border, since this territory had been part of the states of Tamaulipas, Coahuila, Chihuahua, and the territory of Nuevo Mexico. If the US pushes for a Pacific outlet and still manages to get California, Texas might side with the US to "reclaim" these lands. However, the US won't let Texas keep the ridiculous panhandle that reaches up to the 42nd parallel, much likely it will push for a border at the Arkansas River or the 37th parallel.