It can exist, provided it does not initiate force, or engage in fraud.
The most basic role of government is protection from external threat, but if murder by government is never legitimate, how can it carry out that purpose?
The employment of physical force in defense of individual, against for INITIATED by another, is not murder.... Nor is it INITIATED force. It is defensive force, in response to force initiated by an aggressor.
Furthermore, applying the principle of non-initiation of force, must a libertarian government only act to defend its people after it has been attacked?
No. If an intent to initiate force can be demonstrated, state (or the individual) may morally move to restrain it. If a mugger pulls out a pistol and it is clear that his intent to do harm is evident, the victim need not wait until the mugger pulls the trigger, before acting in his own defense. The same would be true of state.
In other words, if the government is aware of a threat, can it be justified in pre-emptively acting against that threat even though no act of aggression has been committed?
No, demonstrated intent is enough.
And, of course, if all taxation is theft, how would a libertarian government fund itself?
In a free society, the necessary functions of state are by definition very limited. As such, the fiscal burden for the administration of state would be small. But such functions as are necessary and proper to the defense of rights (military defense, adjudication, etc..) could be funded by any of several non-aggressive means, including user-fees, lotteries, voluntary contributions, etc.
Income taxation is of course unnecessary, and the nation survived and prospered well without it.