if you read The Gold-Plated Sting that Josf has linked in the Sources.
In reply to the question posed in the title - Are Rothbard and Menger trustworthy? - I believe you would have to know the motives of Rothbard for his endorsement of a 100% reserve system. For Menger, I see nothing with what he said to suggest he should be untrustworthy. Short answer.
Rothbard (part 1):
There are good motives and bad motives that could have been associated with Rothbard's position regarding 100% reserve banking.
The bad motive can be easily understood by reading your link to Lew Rockwell's site - implying he endorsed it with intentions of theft. I don't subscribe to that, however.
More likely, he was trying to reverse the steps that got us into the situation that we are in at the present - provide a method for the public to reclaim part of the bullion through currency exchange. He probably figured this was the most workable solution given the politics of the time.
Rothbard (part 2):
I agree with Rothbard that the labor theory of value as he described it is inconsistent with my understanding of economics - which is more based on fundamental thought rather than academic training.
I am not sure what Spooner and Tucker were arguing for however. I apologize for not taking the time at present to go look it up.
Back to the source - If the fundamental argument is over the labor theory of money as described, Rothbard has a point - though not laid out in the text presented in the above comment.
Rothbard's point can be understood by considering a primitive barter economy. One person produces a product that has some trade value in the market - a trinket for 10 bananas maybe. As primitive craftsmen, some are more or less efficient than others at making the trinkets; however, this has no effect on the market value. The amount of labor that went into the trinkets is irrelevant to how much value was added to the economy.
The amount of value added to the economy in that scenario is the price of the trinket - which is likely set by the most efficient producers.
I don't see anything wrong in Rothbard's premises here. Like I said however, I am not familiar with Spooner and Tucker.
Menger has a good point. Namely, in a society where all desired goods are absolutely abundant, no wealth is required to exist, and therefore, it would not exist. However, this is an idealization and could never occur in a practical sense, IMO.
The first paragraph from Menger you posted in the above comment gives the fundamental source of economic demand.
Like with the trinkets, the availability determines the price, and the price is inversely proportional to market supply. As production becomes more efficient, the supply increases, the price decreases - unless demand can be artificially stimulated to exceed normal levels and keep up with or outpace the supply. Then, the price would remain unchanged or increase, respectively.
I still trust Rothbard and Menger.