Yea, each class has four or five races [that should be] available to it natively. Sith Inquisitor and Sith Warrior should both be creatable as a sith pureblood right from the get go. You can "unlock" the race for other classes (on both Imperial and Republic sides) by reaching level 50 with a character of that race or purchasing the unlock through the Legacy Perks and/or cartel coins.
If you can't make a pureblood inquisitor, it sounds like something's not right. Are you sure your sub is active? It's possible race creation is limited for free accounts.
However, race in SWTOR is pretty much purely aesthetic and affects some story/dialogue elements in small ways if you can look past it as well.
Additionally Korriban is not the most compelling planet. I'd say try to stick it out past level seven and once you reach the planet immediately after it (Dromund Kaas) I think the game will pick up for you. Or you can try a BH or IA and give Hutta a shot, personally I really liked the BH story (and the IA is commonly cited as the best story in the game, mine's only level 20ish) and the Hutta quests I found to be much better than Korriban's.
No experience on the republic side however.