Loss Function for Probability Regression
Given that you are trying to predict a scalar probability value, the cross-entropy formula you listed in the question is only valid if the target variable is discrete. So if your question is "predicts the odds of drawing a matching card from a deck", it will be fine.

The main difference between cross entropy and MSE will be how they penalize wrong predictions. Let's say given a target of 1 but prediction with 0. The cross-entropy is actually undefined in this case, but as the prediction gets closer to 0, the cross-entropy loss gets exponentially larger. On the other hand, your MSE is only 1. Which one is better, it depends on your application, if you want to avoid large margin of errors, it would seem the cross-entropy is more appropriate and vice versa. 

The second and third approach only differs in how they make sure the prediction is within [0, 1], one uses a sigmoid function and another uses a clamp. 
Given you are using a neural network, you should avoid using the clamp function. The clamp function is the same as the identity function within the clamped range, but completely flat outside of it. So the gradient of that function is 1 within the clamp range but 0 outside of it. Because of this, you are more likely to run into the dead neural problem in the same way when people talk about "dead relu".