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In order to design your amplifer with the spec you provided is hard because the S21 and S11 of the HMET or any semiconductor will vary with frequency all I can say at this stage (having done no quantitive analysis) it is very difficult to obtain 10dB gain form a S21 of 15dB which is not same...
Re: RF front end
I think you dont need any matching circuit if the output impedance of the LNA (typically 50Ω) is same as the input impedance of the other LNA and you might want to consider the frequency range you are operating and the microstrip trace impedance connecting the LNA's.
I would first like to know what is the probe impedance i,e if you are using a 50Ω or 1MΩ probe, hopefully if you are using a 1MΩ probe you might want to check your what is the impedance setting on the oscilloscope impedance (50Ω or 1MΩ).
The most likely explanation for blowing the 51Ω resistor...
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