Hadamard transformation and measurement of one qubit
I'm not the best person to answer your question but I'll give it a try.

1st Question:

You do not yet make a measurement after applying the Hadamard Operator on your first qubit. Afterwards say you make a measurement on your 1st qubit, the state you'll end up with is:

$|\psi&gt;_{post-measurement}$ = $1/\sqrt{p_1(m)}$ $\sum_{j} c_{mj} |m&gt;_1 |j&gt;_2$

whereby $m$ is the measurement result of your 1st qubit (either 0 or 1) , $p_1(m)$ is the probability of getting that result on your 1st qubit, $c_{mj}$ the usual coefficient infront of your ket. 

2nd Question:

This will answer how to get ${p_1(m)}$. 
Write out the density matrix of $|\psi&gt;_{pre-measurement}$, then trace out the 2nd qubit. Then apply bra-ket to your reduced density matrix:

${p_1(m)}$ = $&lt;m|_{1}$ $\rho_{1} |m&gt;_{1}$ 

Or equivalently:

${p_1(m)}$ = $\sum_{n}|&lt;m|_1&lt;n|_2 \sum_{ij} c_{ij} |i&gt;_1|j&gt;_2|^2$