The Best "Plausible" outcome for an Independent Texas
Could Texas Survive without Being Annexed?
This first question is Texas' survival. The Lone Star Republic will have to deal with Indians (Apaches) raiding on the western border and Mexicans who don't recognize the treaty which supposedly granted Texas freedom. To survive without drastically altering history (which is of course possible) then Texas will need the help of Britain. Matthais Corvinus is right, such help would probably have come from Britain: indeed, today somewhere in London is a mediocre Tex-Mex restaurant called the Texas Embassy because it occupies the building where the RoT's embassy to Britian stood. The British would offer mediation with the Mexicans, aid, and trade, but probably would have insisted on emancipation. Texas has to agree or face long odds by itself. Most likely legal segregation occurs (if the emancipated slaves aren't sold to the US or exiled to Liberia).
What kind of Texas results between 1845-1860?
We must first explain why Texas wasn't annexed by the US. The more likely scenario than outright Texan refusal to petition for annexation in the first place is that the US continues to refuse annexation due to concerns about slavery. By the time the Texans agree to emancipation to get British aid, opinion is in Texas itself disfavors the Union. The USA meanwhile concentrates on a 54'40" settlement of Oregon Country in 1844. War with Britain may break out, but all we really need is tension.
Britain in enforcing peace between Mexico and Texas also enforces the maximum extent of territory for Texas: the Rio Grande to its source/a line North to the US boundary settled in 1819. Texas has to convince Santa Fe that it is really part of Texas; in OTL, Santa Fe resisted this furiously. This prompts some kind of constitional convetion in Texas. The Republic's Constition up until that point is not federal so admiting New Mexico as a state isn't available as an easy fix. I'd always wondered if given enough time Texas might deviate a bit further from the US' Constitution and invent a new kind of federal structure. I'll hold off on speculation for now; let's presume Texas emerges as a nation still called Texas with the citizens of Santa Fe enthusiastically included.
Without the Mexican American War of 1846-48, the California Gold Rush of 1849 will probably not happen on schedule. In the intervening time, Texas has received the backing of Great Britain and, unlike the US, eliminated slavery. Also, the original Texas Constitution guaranteed certain land rights to settlers; essentially, the Homestead act was part of the Constitution. All of this means that the vast waves of immigrants coming from Europe after 1848 may differ to Texas in greater numbers than OTL. Maybe the British even subsidize starving Irishmen to emigrate to Texas rather than feed the growing numbers of the US; the Irish are happy to go, since there are more Roman Catholics in Texas (i.e. Tejanos, Hispanic Texans).
By the mid-1850s, Texas has a growing imigrant population that is begining to strike out across the Rio Grande to California, hearing of the splendid farming available there. The US has acquired all of Oregon country (Britain is not happy), but the South is all the more incensed because no new slave country has been found. ATL's controversies are even worse for the US. All of this serves to limit the extent to which Western settlers look to Washington when in 1856 or so, gold is discovered in California. In 1858, or so, settlers there rise up against corrupt Mexican officials. They can either chose to pursue life as an independent state, but they will have to fight Mexico City alone: they need some kind of ally. The ATL US is probably not a good candidate. Even if Manifest Destiny sypathies are high, the US will still be riven by slavery debacles. The best bet is Texas or Great Britain, it'd be great to have both.
Mexico will probably have gotten money from Britain in the settlement of the Texas issue, so it doesn't have the debt crises it did OTL. However, Texas will definitely jump at the chance to become a Contiental Republic. Mexico will probably have to be fought, but with the support (not declared) of Britain, volunteer armies, it's even money on the outcome. I'd bet on Texas winning, though it'd be tough. Texas will probably take Baja California and a bit of Sonora, but much more would have to presume either a huge Texas' army or lots of upheval in Mexico.
Questions for Texas after 1860
If the US has a civil war, I'd expect it to stay neutral until Britain does anything. Supporting the US may be a possibitlity in return for some kind of concession out west (maybe Indian territory), but not likely given the descent of the orignial settlers from the South. Supporting the South may be likely because of Realpolitick, but antagonizing Washington is dangerous. I'd still expect to see a Civil War, in anycase, but I'd expect Texas to remain neutral. The biggest effect of the Civil War, IMHO, will be further increased migration to Texas, perhaps some from the former Confederacy due to a desire to escape "Yankee Rule." Once oil is discovered, which I'd expect to see sooner since the RoT will put a heavy emphasis on any kind of drilling technology that allows it to reach water. This will probably quickly have uses if the utility of oil is realized.
The biggest question for this Texas in the later half of the 19th century is how it industrializes. It has all the makings of an extractive economy, relying on agriculture, ranching, and mineral exctraction. British support will probably require Free Trade, which further hampers the development of internal industry. Railroads are the key, however, and Texas has the possibility of developing industry in that arena since it will a shorter route for a transcontintal rail than the US.
In the early 20th century, apart from labor politics, the biggest issue for Texas will be the closing of the frontier: Texas will have relied on land to attract immigrants and homestead rights may have been grandfathered into the RoT Constitution. Eventually, though, the land will run out and something will have to be done.
If Texas doesn't become as expansive, then it becomes something like an American Switzerland, since neutrality will be the only way it can survive and oil will make it wealthy. It will take ingenuity and good leadership for Texas to become more. Whether it will be blessed in those regards is unknowable. I like to think it would be, but I'm biased.