The answer to the first part follows from the law of the iterated expectation for the mean and law of iterated variance for the variance. See here and here for the proofs [I am not sure about the third or higher moments]. For the second part, strict exogenity $E[u_i|X]=0$ implies $E[u_i,X]=0$, but the reverse is not true. To see why this is, the former also implies non-linear (or any function of $X$) independence, e.g., $E[u_i,X^2]=0$.

The answer to the first part follows from the law of the iterated expectation for the mean and law of iterated variance for the variance. See here and here for the proofs [I am not sure about the third or higher moments]. For the second part, strict exogenity $E[u_i|X]=0$ implies $E[u_i,X]=0$, but the reverse is not true. To see why this is, the former also implies non-linear (or any function of $X$) correlation, e.g., $E[u_i,X^2]=0$.

The answer to the first part follows from the law of the iterated expectation. See here for the proof. For the second part, strict exogenity $E[u_i|X]=0$ implies $E[u_i,X]=0$, but the reverse is not true. To see why this is, the former also implies non-linear (or any function of $X$) independence, e.g., $E[u_i,X^2]=0$.

The answer to the first part follows from the law of the iterated expectation for the mean and law of iterated variance for the variance. See here and here for the proofs [I am not sure about the third or higher moments]. For the second part, strict exogenity $E[u_i|X]=0$ implies $E[u_i,X]=0$, but the reverse is not true. To see why this is, the former also implies non-linear (or any function of $X$) independence, e.g., $E[u_i,X^2]=0$.