When do Lasso and linear regression have same solution? When do Lasso and OLS give the same solution when applied independently from each other?
But more importantly for me:
If you first run Lasso and say X* are the features that Lasso has not shrunk to zero (so the features that Lasso has kept in the model with a non-zero coefficient in front of it). And then run OLS with these X* (excluding the features that Lasso has shrunk) will you then get the same results (e.g. same coefficients) as the Lasso?
I have this intuition that you will not get the same, but I cannot fully understand why. I assume the L1 term also has effected the estimation of the coefficients of X*? Loosely speaking that observations were used to shrink and thus it is not same as using all observations to estimate the coefficients of X* in an OLS setting?

The formula for LASSO is
$$\frac{1}{N}||y-X\beta||_2^2+\lambda||\beta||_1$$
Aside from the constant $1/N$, the only difference is the second term $\lambda||\beta||_1$. It is this parameter $\lambda$ which you optimize during lasso and which determines shrinkage. Normally you would choose this using cross validation. Setting $\lambda=0$ (and hence the whole term being equal to 0) you would get the same result as OLS.
