Slavery probably lasts past 1865 in Texas: indeed, a lot of Confederates may end up moving to Texas post civil war [1]. Doubt if it would last past century's end, though: increasing international isolation, British influence (the UK is likely to be a major trading partner), a relatively small pool of slaves to begin with, and the increasing relative importance of oil and cattle versus cotton-growing, all would push for an end to the institution, although some fairly restrictive Black Codes might last well into the 20th. OTOH, given the rather historically fraught realtionship between Texans and Mexicans, we might end the Texan 20th century with a fairly well integrated African-American population and a marginalized hispanic underclass.
As for California, it depends on the timing: with Texas staying out of the US, there is no Mexican-American war, but Texas probably isn't strong enough (yet) in the 1840's to grab California. If the US stays out of the area until the 60's, Texas may have grown strong enough to grab California then while Mexico is in the middle of it's own civil war. The trouble is California gold: when it is found, fortune-hunters from all over will pour in, and the temptation for US or British fillibustering will be quite strong: or once there are enough people there, the Californians might attempt to declare independence from Mexico (it's a long road from mexico city) or do so next time revolution breaks out, in which case we might either get an independent California, one annexed by Texas, or one which votes to join the US. So I'd consider Texan California to exist only in a minority of "Independent Texas" TLs. They might well grab New Mexico and Arizona, however. Whether the Mormons fall to the US, Texas, or manage to pull of independence can be left to the pleasure of the AH maker.
Texas probably remains a democracy. Dubious about that "theocracy" bit: dunno if mid-19th century southerners were really that much more religiously observant than northerners, and the evolution of Texan protestantism isn't necessarily going to be "like OTL, but more so." Texas could be rather _more_ secular than our TL's south by 2006. On the other hand, with the large Hispanic minority, especially if Texas _does_ end up swallowing California, Texas Protestantism might develop a fairly visceral anti-Catholicism that might last longer than it did in the US in our timeline.
Confederate influence probably means US-Texas relations will be prickly for a while, but given their common language, culture, political sysytems, and increasingly important trade ties, Texas and the US probably get along as well as the US and Canada by 2006.
More thoughts later,
Bruce
[1] Unless the lack of Texan annexation somehow butterflies away the Civil war. Frankly, I think it might actually bring it sooner: the South has no potential for future expansion, while the North has all those territories west of the Mississipi and north of Mexico, and they're going to feel more threatened.