How can we be sure of the efficiency of a neural network
OP: Your questions are very good for someone that's just getting started in machine learning.

Have you ensured that the distribution of your training and test dataset are similar? I would try to keep the number of samples per class (label) about equal if possible. For instance, if your training set is severely imbalanced then your prediction algorithm might tend to favor the label that shows up more often.
I think you are on the right track to overfit your model to ensure your neural net architecture, training and whatever else is setup correctly. Are you using regularization? If so, I think you might want to remove that to see if your model can fit to your training dataset. I understand that this goes against what the accepted answer's #2 suggests but this is a useful way to debug your setup
How good are the labels for your dataset? If you have any noise in your labels then that would affect the accuracy of your classifier
You could also try transfer learning if you cannot get more training data
I trained a Forward Neural Net for binary classification and I got an accuracy of 83%, which (I hope)I'm going to improve later by changing parameters in inputs.  But some tests make me feel confused :

My dataset length is 671 so I divide it as 513 train set, 58 Validation set and 100 test set 


When I change the size of my sets (Train, Validation, Test), the accuracy score can decrease to some very low scores like 40% 
The neural net is supposed to learn from the train set but when I test, after the training, with the same Train set and not the Test set, I thought that the model should score 100 %, because he just learned from it, watched it, but it only improves a few with 87% ...

I'm a beginner in ML so I don't know if it's normal or not, I'm just curious and wanna catch all the small things to know, to understand perfectly what I'm doing. I guess it's maybe the normalization of my vectors sets, but I don't know very much about it. I can share you my full code if you want to, but as every neural net, it's quite long but easy to read.