Working ofLSTM with multiple Units - NER
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1) The same input $x_1$ and $h_0$ will be fed to each 100 neurons parallel (independent) and respective outputs will go to next time step. Notice that there are two 100-dimensional vector going to the next time step: $c_1$ and $h_1$, which are denoted by the arrows on the right side in your second figure.

2) This is a complicated question. Firstly, there are usually no one-to-one mapping between a neural and its meaning like neural 1 -> gender. 

3) Those neurons do share information by the matrix multiplication operations. For example, the cell value at time step $t$ is usually calculated as
$$c_t = f_t c_{t-1} + i_t \tanh(W_{xc}x_t+W_{hc}h_{t-1}+b_c)$$
Notice that the hidden value in the previous time step $h_{t-1}$ is multiplied by a matrix $W_{hc}$, therefore each value in $W_{hc}h_{t-1}$ is a linear combination of all the values in $h_{t-1}$.

4) The number of neurons in the LSTM cell, the dimension of hidden vector $h_t$, the dimension of the cell vector $c_t$, and the dimension of the output vector $o_t$, are all the same (in your case, 100). This is known as the "hidden size", "output size" or "number of units/neurons". This is not the same as the dimension of input vector $x_t$, which is usually known as the "input size". 

If you have further questions about the dimension of the LSTM variables, I suggest you to look at the first answer here where the author listed the dimension of all variables and parameters.