It is clear that John and Kate can get together without the intervention of a Terminator, and that they can survive the first such attack, and so reach the events we see in Terminator 3. However, the mission statement of the T-X, which we presume is essentially the same as that of the original terminator sent to this point in time, raises some serious issues.
The problem is that the T-X manages to find at least four of the targets on its list of Conner lieutenants: General Robert Brewster, Jose Barrera, and William and Elizabeth Anderson. If it kills them, they never matter and are not on the list; if they are not on the list, they matter and will survive so as to be listed. It is at least potentially another grandfather paradox. We need to determine how these four people wind up on the hit list despite having been found by the T-X.
General Brewster is easiest to resolve. It is unlikely that he would have survived in any event, and is probably listed as a target because he is a threat independently of any connection he has to John Conner: he oversaw the launching of SkyNet, and has the best chance to know how to fight it. SkyNet lists its creator because he might be its destroyer.
Although upon hearing shots and commotion downstairs, Elizabeth Anderson might have escaped through a window and vanished so as to survive, it seems fairly certain that William Anderson, her brother, is dead. If this is the correct William Anderson, this destroys time. However, the T-X does nothing more in confirming his identity than ask if he is William Anderson, brother of Elizabeth, which he confirms. The original T-800 killed two Sarah Conners before finding the right one; it is within the realm of possibility that this is not that William Anderson. It is even possible that John Conner's lieutenants in the future took names of those already dead in the past, to hide their trail.
That solution does not work for Jose Barrera, however. The T-X confirms his identity with a retinal scan, apparently against information brought from the future. The information for that verification can only exist if Jose Barrera survives. However, the T-X did not confirm that kill; Jose was behind the window at the drive-thru, and certainly fell back, probably down, on the first shot. Emergency services would be called, and he would be rushed to a hospital. Assuming he survived, his parents, probably Mexican immigrants, may well have decided the city was too dangerous, and taken him elsewhere.
It could be argued that an infinity loop is avoided because the T-X will transmit its information to SkyNet, and so SkyNet will have the list from the moment of its creation. However, the T-X never interfaces directly with SkyNet. We do know, though, that the T-X accessed a networked computer to find information about those on its list. It might also have stored the list there for SkyNet to find once it became active. Since SkyNet takes over all interconnected computer networks, it would ultimately find the list. This particular form of predestination paradox, in which information passes from the future, is easily resolved, as we saw in Star Trek IV: The Voyage Home with transparent aluminum and again in the 2009 Star Trek in connection with Mr. Scott's forumula for transwarp teleportation.
As to the three other deaths--Scott, Betsy, and the woman in the car--they would have been in the city when the bombs fell, and so would not have lived more than another few days anyway. Although these are not the best solutions, these events could happen without necessarily destroying time.
Which finally brings us to the beginning of the latest film, Terminator Salvation.
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