Measurements are never instantaneous, only idealizations of actual measurements are modeled as instantaneous.
Measuring one particle in a entangled pair destroys the entanglement, after a measurement they are no longer entangled.
Doing repeated measurements quicker and quicker can exert a quantum Zeno effect where the no-measurement evolution away from a measurement eigenstate is difficult to achieve because of the repeated measurements.
What happens in a measurement is you split the state into eigenstates, so an entangled state become not entangled but get the correlation that was encoded in the entanglement. After that first measurement they have that correlation, but are no longer entangled, you can further split the one and the other just is what it is (the same thing) for each of the splits after the first measurement.
And whenever you do a measurement, you have to say exactly what you are measuring, different measurements are incompatible, so you can't do them all, and doing one first and then the other might give different results than doing the other first and then doing the one. So just talking about "measuring" one is too vague. Even a very simple measurement such as spin that only produces two results (say $\pm \hbar/2$) still has many choices about which axis to measure it, and you can't measure them all and the order you measure them matters when you do measure different ones.