I find the "MLR X" format somewhat confusing, so I won't use it here. They basically refer to the conditions under which the Gauss-Markov Theorem applies, i.e. the conditions under which the OLS estimator is BLUE (best linear unbiased estimator). The most important ones are: 1) Regressors must be exogenous, i.e. $E[u|X]=0$, and 2) errors must be homoskedastic, i.e. $Cov[u]=\sigma^2I$ where $I$ is the identity matrix. I tried to break down your post into 2 main questions:
Question 1: How can we have unbiased estimators with heteroskedasticity? Answer: unbiasedness has nothing to do with the variance and is not the result of the Gauss-Markov theorem. For unbiasedness it exogeneity suffices, i.e. $E[u|X]=0$. Proof: $$E[\hat \beta_{OLS}|X]=E[(X^TX)^{-1}X^Ty|X] \\ = E[(X^TX)^{-1}X^T(X\beta + u)|X] \\ =(X^TX)^{-1}(X^TX)\beta \underbrace{E[u|X]}_{=0}=\beta$$
Notice how the variance of $u$ is not imporant here, i.e. it could be that $Var[u_i]=\sigma^2_i$ with a different $\sigma^2_i$ each time (i.e. heteroskedasticity). Further notice that exogeneity implies no correlation, i.e. $u_i$ can not be autocorrelated.
Question 2: Can the Gauss-Makrov Theorem still hold under heteroskedasticity? Under heteroskedasticity, the Gauss-Markov Theorem does not apply anymore, i.e. the OLS estimator is not BLUE anymore. It still a LUE, i.e. linear and unbiased, but not best anymore. Here "best" refers to the estimator with the lowest variance. The GLS (generalized least squares) estimator is BLUE now instead, if $\Omega$ is any variance-covariance matrix: $\beta_{GLS}=(X^T\Omega^{-1} X)^{-1}X^T \Omega^{-1}X$. Proof is somewhat lengthy, but not hard. Essentially it boils down to proving that $Var[\beta_{OLS}]-Var[\beta_{GLS}]\ge0$ for any $\Omega$.