Ground state of a SHO has a gaussian wave function in which the average position is calculated in a standard way yielding x_avg = 0 as result (of course the center of the gaussian is suposed to be at x = 0 too).
The energy can be calculated with the n quantum number, so that n = 0 corresponds to the energy of [itex] 1/2 \hbar \omega [/itex]. Answering your sencond question, n=0 corresponds to groud state.
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